Alta - Chapter 4 - Discrete Random Variables - Part 1

A random sample of college students majoring in cinematography were surveyed about their movie habits. One question in the survey asked, "How many documentaries have you watched in the past month?" The table below represents the probability density function for the random variable X, the number of documentaries watched by cinematography majors in the past month. Find the standard deviation of X. Round the final answer to two decimal places.

1.50

The Stomping Elephants volleyball team plays 30 matches in a week-long tournament. On average, they win 4 out of every 6 matches. What is the mean for the number of matches that they win in the tournament?

20 The mean of a binomial distribution is the product of p, the probability of a success, and n, the total number of repeated trials or events. mean = μ=np In this scenario, the number of trials is 30 (the total number of matches played), which is n. The probability of a success (winning a match) is 46. So, the mean of the binomial distribution B(30,46) is: μ=(30)(46)=20

65% of the people in Missouri pass the driver's test on the first attempt. A group of 5 people took the test. What is the probability that less than 3 in the group pass their driver's tests in their first attempt? Round your answer to three decimal places. Remember: 65% = 0.65.

65% of the people in Missouri pass the driver's test on the first attempt. A group of 5 people took the test. What is the probability that less than 3 in the group pass their driver's tests in their first attempt? Round your answer to three decimal places. Remember: 65% = 0.65.

On average, Roger has noticed that 9 runners pass by his house each day. What is the probability that exactly 5 runners will pass his house in a day? (Answers are rounded to the thousandths digit.)

Correct answer: 0.061 The time interval of interest is a day. The average number is 9 per day, so if X is the number of runners that Roger sees in a day, then X∼Po(λ) where λ=9. We want to know the probability of 5 runners in a day, which means x=5. According to the formula, we find P(X=x)=λxe−λx!=95e−95!≈0.061

Nick gets an average of 18 calls during his 8 hour work day. What is the probability that Nick will get exactly 10 calls in a 5 hour portion of his work day? (Round your answer to three decimal places.)

Correct answers:$0.116$0.116​ The time interval of interest is 5 hours. There is an average of 18 calls per 8 hours or 1885=454 calls per 5 hours. The probability can be found using the Poisson distribution with parameter λ=454. Let X be the number of calls that are received in the 5 hour time period. According to the formula, we find: P(X=x)=λxe−λx! P(X=10)=(454)10e−45410! ≈0.116

Consider how the following scenario could be modeled with a binomial distribution, and answer the question that follows. 54.4% of tickets sold to a movie are sold with a popcorn coupon, and 45.6% are not. You want to calculate the probability of selling exactly 6 tickets with popcorn coupons out of 10 total tickets (or 6 successes in 10 trials). What value should you use for the parameter p?

Correct answers:$0.544$0.544​ The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a movie ticket with a popcorn coupon, so p=0.544.

A softball pitcher has a 0.487 probability of throwing a strike for each pitch. If the softball pitcher throws 29 pitches, what is the probability that no more than 14 of them are strikes? Insert the correct symbol to represent this probability.

This probability could be represented by P(X≤14) or P(X<15)

Bernoulli trial

an experiment with a set of identical trials each with only two possible outcomes.Bernoulli trial is also known as a Binomial trial

Which are properties of a random variable? Select all that apply.

Random variables are denoted with an upper case letter. Random variables are not represented by numerical values. Notation for a random variable is an upper case letter. These variables are usually represented with words, and do not have numerical values. Recall that discrete data are data that you can count, whether it is random variable or not.

Leah's answering machine receives about 6 telephone calls between 8 a.m. and 10 a.m. What is the average number of calls Leah receives in 15 minutes?

Since the mean, μ, is equal to np, you need to determine the values of n (the average over the whole interval) and p (the ratio of the chosen interval to the entire interval). The average number of phone calls over the entire interval is 6. So, n=6. The entire time interval is from 8 a.m. and 10 a.m., which is 2 hours, or 120 minutes. The interval you are looking at is just a 15 minute interval. So, p=15120=18.(*Note: Be sure to compare the same units!) Substitute the values for n and p in the equation to find the mean. μ=np=(6)(18)=0.75 So, the mean is 0.75, which means that Leah should expect 0.75 telephone calls in 15 minutes.

You are flipping three fair coins, and want to know the possible outcomes. Let the random variable X = the number of heads you get when you toss three fair coins. The sample space (possible outcomes) for the toss of three fair coins is TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. (H = heads; T = tails)

So, x = 0,1,2,3, because when you toss three coins, you have the possibility of getting 0 heads, 1 head, 2 heads, or 3 heads.

It has been stated that about 41% of adult workers have a high school diploma, but do not pursue higher education. If 20 adult workers are randomly selected, what is the mean, variance, and standard deviation of X∼B(20,0.41)?

The notation tells you that X is a random variable with a binomial distribution, where X= the number of workers who have a high school diploma but do not pursue higher education. To find the mean, (μ=np), you need to know the values of n and p. From the problem, and the notation, X∼B(20,0.41), you know that the number of trials, n, is 20, and that p=0.41 (from the 41% of adult workers with a high school diploma). mean (μ) =(20)(0.41)=8.2 This means that in a group of 20 adult workers, on average, there will be approximately 8.2 (or 8) of those workers who have a high school diploma, but did not pursue higher education. To find the variance, (σ2=npq), you need to know the values of n, p, q. Because there are only two options ("higher education" or "no higher education"), and because this is a binomial distribution, you know that p+q=1. Solving 0.41+q=1 for q, we find that q=0.59. σ2 (variance) =npq=(20)(0.41)(0.59)=4.838 The standard deviation is just the square root of the variance. σ (stdev) = σ2−−√≈2.20

discrete probability density function

We summarize the probability that a discrete random variable X takes on certain values of x by way of a discrete probability density function. A discrete probability density function describes the relative likelihood for a discrete random variable to take on a given value. The probability density function is usually shown in a table, with one column labeled 'x,' showing the specific values the random variable X can have, and one column labled "P(X = x)," which shows the probability of the random variable X taking on that value.

The parameters of the binomial distribution:

X (the random variable) = the number of successes p = the probability of a success q = the probability of a failure n = the number of independent trials μ (mean) = np σ2 (variance) = npq σ (standard deviation) = npq−−−√

Binomial distribution

a common discrete probability function that includes a fixed number of identical trials in an experiment where the only possible outcomes are success and failureBinomial distribution is sometimes known as Bernoulli distribution

An automobile manufacturer is concerned about the possibility of a fault in the engine of a particular model of one of its cars. This fault could potentially cause the engine to fail to start. The average number of cars per year that experience this fault is 3 cars. If the probability that 8 or more cars experience this fault in a year is greater than 0.05, the company will issue a recall. Will the company issue a recall for this engine fault? Use Excel to find the probability.

"The company will not issue a recall because the probability is less than 0.05." Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 8 occurrences or more for a year. This is the complement of the probability of 7 occurrences or fewer. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 7 and 3, in that order. In step 3 of the explanation, changed to "Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.988095, which is 0.988 rounded to three decimal places. To find the probability of 8 or more cars experiencing the fault, subtract this probability from 1. The probability is 1−0.988=0.012. Since this probability is less than 0.05, the manufacturer will not issue a recall.

In a recent baseball season, Ron was hit by pitches 21 times in 602 plate appearances during the regular season. Assume that the probability that Ron gets hit by a pitch is the same in the playoffs as it is during the regular season. In the first playoff series, Ron has 23 plate appearances. What is the probability that Ron will get hit by a pitch exactly once? Use Excel to find the probability. Round your answer to three decimal places.

$0.367$0.367​ Note that this is a binomial probability. In this case, we want to find the probability of 1 success, where a success is getting hit by a pitch. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 1, 23, and 21/602, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.367369, which is 0.367 rounded to three decimal places

In a recent baseball season, Ron was hit by pitches 24 times in 633 plate appearances during the regular season. Assume that the probability that Ron gets hit by a pitch is the same in the playoffs as it is during the regular season. What is the probability that Ron gets his first hit-by-pitch of the playoffs in his first 16 plate appearances? Use Excel to find the probability. Round your answer to three decimal places

$0.461$0.461​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be Ron getting hit by a pitch. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 16 trials, there are at most 15 failures before the first success. Thus, enter 15, 1, and 24/633, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.461213, which is 0.461 rounded to three decimal places.

A database system assigns a 32-character ID to each record, where each character is either a number from 0 to 9 or a letter from A to F. Assume that each number or letter being selected is equally likely. Find the probability that at least 20 characters in the ID are numbers. Use Excel to find the probability. Round your answer to three decimal places.

$0.578$0.578​ Note that this is a binomial probability. In this case, we want to find the probability of 20 to 32 successes, inclusive, where a success is a character being a number. This is the complement of the probability of 0 to 19 successes, inclusive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 19, 32, and 0.625, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.42192, which is 0.422 rounded to three decimal places. To find the probability of 20 to 32 successes, subtract this probability from 1. The probability is 1−0.422=0.578.

A basketball player has a 0.604 probability of making a three-point shot. What is the probability that it takes the player 5 tries to make a three-point shot? Round your answer to three decimals

0.015

A restaurant gets an average of 10 phone orders in a 2 hour time period. In order to find the probability that the restaurant will get exactly 0 phone orders in a 20 minute period using the Poisson distribution, what is the time interval of interest?

Correct answers:$20\ \min$20 min​ The time interval of interest is the fixed time period for which the probability of an event is being sought. In this case, the time interval of interest is 20 minutes.

Alex wants to test the reliability of "lie detector tests," or polygraph tests. He performs a polygraph test on a random sample of 12 individuals. If there is more than a 50% chance that the tests result in no false positives (that is, the test does not result in a true statement being recorded as a lie), Alex will conclude that the tests are reliable. If the probability of a lie detector test resulting in a false positive is 5.5%, what will Alex conclude? Use Excel to find the probability, rounding to three decimal places.

Alex will conclude that the test is reliable since the probability of no false positives is greater than 0.5. Note that this is a binomial probability. In this case, we want to find the probability of 0 successes, where a success is getting a false positive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 0, 12, and 0.055, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.507203, which is 0.507 rounded to three decimal places. This probability is greater than 0.5, so Alex will conclude that the test is reliable.

Correct answer: The company will not issue a recall because the probability is less than 0.05. Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 8 occurrences or more for a year. This is the complement of the probability of 7 occurrences or fewer. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 7 and 3, in that order. In step 3 of the explanation, changed to "Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.988095, which is 0.988 rounded to three decimal places. To find the probability of 8 or more cars experiencing the fault, subtract this probability from 1. The probability is 1−0.988=0.012. Since this probability is less than 0.05, the manufacturer will not issue a recall.

An automobile manufacturer is concerned about the possibility of a fault in the engine of a particular model of one of its cars. This fault could potentially cause the engine to fail to start. The average number of cars per year that experience this fault is 3 cars. If the probability that 8 or more cars experience this fault in a year is greater than 0.05, the company will issue a recall. Will the company issue a recall for this engine fault? Use Excel to find the probability.

Bernoulli Trial

Any experiment that has just two possible outcomes ("success" and "failure") for each trial, and where the n trials are independent and repeated using identical conditions and n=1 is called a Bernoulli Trial. This type of trial is named after Jacob Bernoulli who, in the late 1600s, studied them extensively. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.

A weighted coin has a 0.564 probability of landing on heads. What is the probability that it takes 4 flips for the first head to occur? Round the final answer to three decimal places.

Correct answers:$0.047$0.047​ This is an example of a geometric distribution with success probability p=0.564. Find q by q=1−p=1−0.564=0.436 Then take the equationP(X=k)=q(k−1)⋅pWhere k=4:P(X=4)=0.436(4−1)⋅0.564P(X=4)=0.436(3)⋅0.564P(X=4)=0.047 By using a calculator or computer, the probability that it takes 4 flips for the first head to occur is P(X=4)=geometpdf(0.564,4)=0.047.

Burt, a football quarterback, has a pass completion percentage of 55.2%. If the probability that Burt will need 8 or more pass attempts to complete his first pass in a game is at least 0.01, then Burt will be benched and replaced by a backup. Will Burt be benched? Use Excel to find the probability, rounding to three decimal places.

Burt will not be benched since the probability is less than 0.01. The probability that Burt will need eight or more attempts to complete his first pass is the complement of the probability that he will need at most seven attempts. Use Excel to find the probability. Let a success be Burt completing a pass. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most seven trials, there are at most six failures before the first success. Thus, enter 6, 1, and 0.552, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.996378, which is 0.996 rounded to three decimal places. The complement of this probability is 1−0.996=0.004, which is less than 0.01. So Burt will not be benched.

The National Coffee Association claimed that in 2010, Americans drank an average of 3 cups of coffee per day. Let X = the number of cups of coffee Joe drinks in a day. The random variable X has a Poisson distribution: X∼Po(3) . What is the probability that he drinks 2 or fewer cups of coffee in a day? (Answers are rounded to the thousandths digit.)

Correct answer: 0.423 The average is 3, so the parameter is λ=3. Remember that the formula is P(X=x)=λxe−λx! The probability of drinking 2 or fewer cups isP(X≤2)=P(X=0)+P(X=1)+P(X=2)=30e−30!+31e−31!+32e−32!≈0.050+0.149+0.224≈0.423

The table below shows a probability density function for a discrete random variable X. What is the probability that X is 2?

Correct answer: 1/12 Find the row where x = 2 (the third row in the table). The probability that X is 2 can be found in the righthand column of the table, in this row. So, the probability that X is 2 is 1/12.

An average person can type 40 words per minute. In order to find the probability that a person will type more than 100 words in 3 minutes using the Poisson distribution, what is the average number of words per 3 minutes?

Correct answer: 120 words Since the mean, μ, is equal to np, you need to determine the values of n (the average over the whole interval) and p (the ratio of the chosen interval to the entire interval). The average number of words per minute is given as 40. So, n=40. The entire time interval the question asks for is 3 minutes. This interval is 3 times as long as the original. So, p=3. Substitute the values for n and p in the equation to find the mean. μ=np=(40)(3)=120 So, the mean is 120 words per 3 minutes

Using the same scenario, what is the standard deviation for the number of matches that they win in the tournament? The Stomping Elephants volleyball team plays 30 matches in a week-long tournament. On average, they win 4 out of every 6 matches.

Correct answer: 6.67−−−−√ The standard deviation of a binomial distribution is the square root of the variance. The variance is the product of n, the total number of repeated trials or events, p, the probability of a success, and q, which is (1−p). variance = σ2=np(1−p) standard deviation = variance−−−−−−−√ In this scenario, the number of trials is 30 (the total number of matches played), which is n. The probability of a success (winning a match) is 46. So, the value for (1−p) is 26. The variance is the product of these values. variance = σ2=(30)(46)(26) So the variance is approximately 6.67, and the standard deviation is the square root of that, 6.67−−−−√.

An insurance company is assessing the risk of selling flood insurance. Under the company's policies, if a customer lives in an area where they have a greater than 5% chance of experiencing 4 or more floods in 10 years, the customer must pay a higher premium. Drew lives in an area where the average number of floods per 10 years is 1.6 floods. Will Drew have to pay the higher premium for flood insurance? Use Excel to find the probability. Select the correct

Correct answer: Drew will have to pay the higher premium because the probability is greater than 0.05. Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 4 occurrences or more for 10 years. This is the complement of the probability of 3 occurrences or fewer. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 3 and then 1.6, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.921187, which is 0.921 rounded to three decimal places. To find the probability of 4 or more floods, subtract this probability from 1. The probability is 1−0.921=0.079. Since this probability is greater than 0.05, Drew will have to pay the higher premium.

Bryan is performing data entry for an insurance company by taking handwritten records and entering them into the database. Thus far, 1.8% of the handwritten records have had missing data that prevented them from being entered into the database. If the probability that the first record with missing data is found within the first five records is greater than 5%, Bryan will stop data entry and ask for more information. Will Bryan stop entering data? Use Excel to find the probability, rounding to three decimal places.

Correct answer: Yes, because the probability is greater than 5%. Use Excel to find the probability. Note that this is a cumulative probability. Let a success be an incomplete record. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most five trials, there are at most four failures before the first success. Thus, enter 4, 1, and 0.018, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.086818, which is 0.087 rounded to three decimal places. This is greater than 0.05, so Bryan will stop data entry.

A computer graphics card manufacturer is testing an improvement to its production process. If a sample of 100 graphics cards manufactured using the new process has a less than 10% chance of having 3 or more defective graphics cards, then the manufacturer will switch to the new process. Otherwise, the manufacturer will stay with its existing process. If the probability of a defective graphics card using the new process is 0.9%, will the manufacturer switch to the new production process

Correct answer: Yes, because the probability of having 3 or more defective graphics cards is less than 0.10. Note that this is a cumulative binomial probability. In this case, we want to find the probability of 3 or more successes, inclusive, where a success is one of the graphics cards being defective. The probability of having 2 or fewer defective graphics cards is the complement of the probability of having 3 or more defective graphics cards. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 2, 100, and 0.009, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.937964, which is 0.938 rounded to three decimal places. To find the probability of having 3 or more defective graphics cards, subtract this probability from 1. The probability of having 3 or more defective graphics cards is 1−0.938=0.062, which is less than 0.10. So, the manufacturer will switch to the new process.

Identify the parameters p and n in the following binomial distribution scenario. Jack, a bowler, has a 0.38 probability of throwing a strike and a 0.62 probability of not throwing a strike. If Jack bowls 20 times, he wants to know the probability that he throws more than 10 strikes. (Consider a strike a success in the binomial distribution.)

Correct answer: p=0.38,n=20 In a binomial distribution, there are only two possible outcomes. p denotes the probability of the event or trial resulting in a success. In this scenario, it would be the probability of Jack bowling a strike, which is 0.38. The total number of repeated and identical events or trials is denoted by n. In this scenario, Jack bowls a total of 20 times, so n=20.

Jackie is practicing free throws after basketball practice. She makes a free throw shot with probability 0.7. She takes 20 shots. We say that making a shot is a success. What are p, q, and n in this context?

Correct answer: p=0.7, q=0.3, n=20 Remember that p is the probability of success, which is the probability of making a shot, 0.7. The probability of failure is q=1−p=0.3. n is the number of repetitions, which is 20.

According to a Gallup poll, 60% of American adults prefer saving over spending. Let X= the number of American adults out of a random sample of 50 who prefer saving to spending. What is the mean (μ) and standard deviation (σ) of X?

Correct answer: μ=30 and σ≈3.46 Remember that the mean μ is given by the formula μ=np. This should make sense because you can think of p as the fraction of the sample, on average, that will be a success. In this case p=0.6 because we think of a success as someone who prefers saving over spending. n is the size of the sample, 50. So μ=(50)(0.6)=30 Standard deviation is given by the formula σ=npq−−−√. As above, n=50 and p=0.6. Remember that p+q=1, so solving for q we find that q=1−p=0.4. So σ=(50)(0.6)(0.4)−−−−−−−−−−−√=12−−√≈3.46

During a bowling league tournament, the number of times that teams scored a strike every ten minutes was recorded by a scorekeeper. The table below represents the probability density function for the random variable X, the number of strikes every ten minutes. Find the standard deviation of X. Round the final answer to two decimal places.

Correct answers: std=2.81 strikes​ Here's the filled out table. x0348P(X=x)16161313x⋅P(X=x)0124383(x−μ)2P(X=x)3.3750.3750.08334.0833 Note that the mean (also called expected value) μ is the sum of the entries in the third column:μ=12+43+83=4.5This value is used to compute the values in the fourth column. For example, the first entry is(0−4.5)2⋅16=3.375Summing the values in the fourth column gives the variance:3.375+0.375+0.0833+4.0833=7.9166Taking the square root of the variance gives the standard deviation:σ=7.9166−−−−−√≈ 2.81

At a local grocery store, customers purchase an average of 2,434 boxes of breakfast cereal each week. Find the probability that customers purchased no more than 2,300 boxes of cereal in a randomly selected week. Use Excel to find the probability. Round our answer to three decimal places.

Correct answers:$0.003$0.003​ Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 0 to 2,300 occurrences for a week, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 2300 and 2434, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.003187, which is 0.003 rounded to three decimal places.

A soda bottling company's manufacturing process is calibrated so that 99% of bottles are filled to within specifications, while 1% is not within specification. Every hour, 12 random bottles are taken from the assembly line and tested. If 2 or more bottles in the sample are not within specification, the assembly line is shut down for recalibration. What is the probability that the assembly line will be shut down, given that it is actually calibrated correctly? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.006$0.006​ Note that this is a binomial probability. In this case, we want to find the probability of 2 or more successes, where a success is a bottle not filled to within specifications. This is the complement of the probability of 0 to 1 successes, inclusive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 1, 12, and 0.01, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.993825, which is 0.994 rounded to three decimal places. To find the probability of 2 or more successes, subtract the probability of 0 or 1 successes from 1. This probability is 1−0.994=0.006.

A company manufactures graphics cards for computers. About 1.5% of the graphics cards are rejected by a quality check before they are sent to customers. What is the probability that the 15th graphics card tested is the first to be rejected? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.012$0.012​ Use Excel to find the probability. Let a success be a graphics card being rejected by the quality check. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of 15 trials, there are 14 failures before the first success. Thus, enter 14, 1, and 0.015, in that order. In the entry for a cumulative probability, enter 0 or leave it blank because this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.012139, which is 0.012 rounded to three decimal places.

A marksman has a probability of 0.624 for hitting a certain long range target. What is the probability that it takes 4 shots for the marksman to hit the long range target? Round your answer to three decimals.

Correct answers:$0.033$0.033​ This is an example of a geometric distribution with success probability p=0.624. Find q by q=1−p=1−0.624=0.376 Then take the equationP(X=k)=q(k−1)⋅pWhere k=4:P(X=4)=0.376(4−1)⋅0.624P(X=4)=0.376(3)⋅0.624P(X=4)≈0.033 By using a calculator or computer, the probability that it takes 4 shots for the marksman to hit the long range target is P(X=4)=geometpdf(0.624,4)≈0.033.

In a recent basketball season, a team scored 107.7 points per 100 possessions. Find the probability that, in a randomly selected interval of 100 possessions, the team scored exactly 112 points. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.035$0.035​ Note that this is a Poisson probability. In this case, we want to find the probability of the basketball team scoring exactly 112 points for100 possessions. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First, press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 112 and 107.7, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.034609, which is 0.035 rounded to three decimal places.

Chris is trying to get his driver's license. Assume that the probability of someone passing the driver's test is 78%. What is the probability that Chris succeeds on his third attempt? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.038$0.038​ Use Excel to find the probability. Let a success be Chris passing the test. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of three trials, there are two failures before the first success. Thus, enter 2, 1, and 0.78, in that order. In the entry for a cumulative probability, enter 0 or leave it blank because this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.037752, which is 0.038 rounded to three decimal places.

A basketball player has a 0.498 probability of making a three-point shot. What is the probability that it takes the player 4 tries to make a three-point shot? (Round your answer to three decimals if necessary.)

Correct answers:$0.063$0.063​ This is an example of a geometric distribution with success probability p=0.498. Find q by q=1−p=1−0.498=0.502 Then take the equationP(X=k)=q(k−1)⋅pWhere k=4:P(X=4)=0.502(4−1)⋅0.498P(X=4)=0.502(3)⋅0.498P(X=4)=0.063 By using a calculator or computer, the probability that it takes the player 4 tries to make a three-point shot is P(X=4)=geometpdf(0.498,4)=0.063.

A stock exchange computes that there is an average of 2.7 initial public offerings (IPOs) per week on the exchange. Find the probability that there are 5 IPOs in a randomly selected week. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.080$0.080​ Note that this is a Poisson probability. In this case, we want to find the probability of 5 occurrences for a week. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 5 and 2.7, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.08036, which is 0.080 rounded to three decimal places.

You play a casino game until you win for the first time. On average, you win for the first time on the 11th game played. What is the probability of winning the game? Round your answer to three decimal places.

Correct answers:$0.091$0.091​ This is an example of a geometric distribution. If p represents the probability of success, then the expected number of trials is n=1p. Solving this equation for p shows that p=1n. In this case n=11, so p=1/11=0.091.

You purchase boxes of cereal until you obtain one with the collector's toy you want. If, on average, you get the toy you want in every 11th cereal box, what is the probability of getting the toy you want in any given cereal box? (Round your answer to three decimals if necessary.)

Correct answers:$0.091$0.091​ This is an example of a geometric distribution. If p represents the probability of success, then the expected number of trials is n=1p. Solving this equation for p shows that p=1n. In this case n=11, so p=111=0.091.

A restaurant gets an average of 14 calls in a 2 hour time period. What is the probability that the restaurant will get at most 2 calls in a 45 minute period? Round the final answer to three decimal places.

Correct answers:$0.105$0.105​ The time interval of interest is 45 minutes. There is an average of 14 calls per 2 hours or 1412045=214 calls per 45 minutes. The probability can be found using the Poisson distribution with parameter λ=214. Let X be the number of calls that are received in the 45 minute time period. According to the formula, we find: P(X=x)=λxe−λx! P(X≤2)=P(X=0)+P(X=1)+P(X=2) P(X≤2)=(214)0e−2140!+(214)1e−2141!+(214)2e−2142! 0.005+0.028+0.072 ≈0.105

Sarah is proofreading an article. From previous experience, she expects an average of 5.3 typographical errors per 100 words. Find the probability that a randomly selected section of 100 words has 7 typographical errors. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.116$0.116​ Note that this is a Poisson probability. In this case, we want to find the probability of 7 occurrences for 100 words. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 7 and 5.3, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.116343, which is 0.116 rounded to three decimal places.

Stephanie gets an average of 11 calls during her 8 hour work day. What is the probability that Stephanie will get more than 4 calls in a 2 hour portion of her work day? (Round your answer to three decimal places.)

Correct answers:$0.144$0.144​ The time interval of interest is 2 hours. There is an average of 11 calls per 8 hours or 1182=114 calls per 2 hours. The probability can be found using the Poisson distribution with parameter λ=114. Let X be the number of calls that are received in the 2 hour time period. According to the formula, we find: P(X=x)=λxe−λx! Since it is easier to find X equal to or less than 4, we use the compliment:P(X>4)=1−(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4))P(X>4)=1−((114)0e−1140!+(114)1e−1141!+(114)2e−1142!+(114)3e−1143!+(114)4e−1144!)1−(0.064+0.176+0.242+0.222+0.152)≈0.144

In baseball, the statistic Walks plus Hits per Inning Pitched (WHIP) measures the average number of hits and walks allowed by a pitcher per inning. In a recent season, Burt recorded a WHIP of 1.315. Find the probability that, in a randomly selected inning, Burt allowed a total of 3 or more walks and hits. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.146$0.146​ Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 3 or more occurrences. This is the complement of the probability of 0 to 2 occurrences for one inning, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 2 and 1.315, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.853644, which is 0.854 rounded to three decimal places. To find the probability of 3 or more walks plus hits, subtract this probability from 1. The probability is 1−0.854=0.146.

Jerry has a bird feeder which is visited by an average of 14 birds every 2 hours during daylight hours. What is the probability that the bird feeder will be visited by more than 2 birds in a 25 minute period during daylight hours? Be sure to round your answer to three significant figures.

Correct answers:$0.558$0.558​ The time interval of interest is 25 minutes. There is an average of 14 birds per 2 hours or 1412025=3512 birds per 25 minutes. The probability can be found using the Poisson distribution with parameter λ=3512. Let X be the number of birds that visit the feeder in the 25 minute time period. According to the formula, we find: P(X=x)=λxe−λx! Since it is easier to find X equal to or less than 2, we use the compliment:P(X>2)=1−(P(X=0)+P(X=1)+P(X=2))P(X>2)=1−((3512)0e−35120!+(3512)1e−35121!+(3512)2e−35122!)1−(0.054+0.158+0.230)≈0.558

In a recent basketball season, Jenny sunk a three-point shot once in every 3.5 attempts, on average. Assume that this probability did not change going into the next season. What is the probability that Jenny sinks her first three-point shot on her third attempt of the season? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.146$0.146​ Use Excel to find the probability. Note that this is not a cumulative probability. Let a success be Jenny sinking a three-point shot. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of three trials, there are 2 failures before the first success. Thus, enter 2, 1, and 1/3.5, in that order. In the entry for a cumulative probability, enter 0 since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.145773, which is 0.146 rounded to three decimal places.

Question On average, Andrew has noticed that 14 trains pass by his house daily (24 hours) on the nearby train tracks. What is the probability that exactly 6 trains will pass his house in a 12 hour time period? (Round your answer to three decimal places.)

Correct answers:$0.149$0.149​ The time interval of interest is 12 hours. There is an average of 14 trains per 24 hours or 142412=7 trains per 12 hours. The probability can be found using the Poisson distribution with parameter λ=7. Let X be the number of trains that pass in the 12 hour time period. According to the formula, we find: P(X=x)=λxe−λx! P(X=6)=76e−76! ≈0.149

In a large city's recent mayoral election, 126,519 out of 283,143 registered voters actually turned out to vote. If 20 registered voters are randomly selected, find the probability that exactly 8 of them voted in the mayoral election. Use Excel to find the probability, Round your answer to three decimal places.

Correct answers:$0.164$0.164​ Note that this is a binomial probability. In this case, we want to find the probability of exactly 8 successes, where a success is a registered voter turning out to vote. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 8, 20, and 126519/283143, in that order. In the entry for a cumulative probability, enter 0 since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.164322, which is 0.164 rounded to three decimal places

The state department of transportation is coordinating road crews to fix potholes after a particularly snowy winter. Initial estimates gave an average of 7.8 potholes per mile of highway. Find the probability that there are 11 or more potholes in a randomly selected one-mile stretch of highway. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.165$0.165​ Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 11 or more occurrences for a one-mile stretch. This is the complement of the probability of 0 to 10 occurrences for a one-mile stretch, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 10 and 7.8, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.83523, which is 0.835 rounded to three decimal places. To find the probability of 11 or more potholes, subtract this probability from 1. The probability is 1−0.835=0.165.

A certain cold remedy has an 88% rate of success of reducing symptoms within 24 hours. Find the probability that in a random sample of 45 people who took the remedy, 40 of them had a reduction of symptoms within a day. Round your answer to three decimal places.

Correct answers:$0.183$0.183​ Note that this is a binomial probability. In this case, we want to find the probability of 40 successes, where a success is a reduction of symptoms. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 40, 45, and 0.88, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.18289, which is 0.183 rounded to three decimal places.

Kelsey, a basketball player, hits 3-point shots on 38.1% of her attempts. If she takes 14 attempts at 3-point shots in a game, what is the probability that she hits exactly six of them? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.198$0.198​ Note that this is a binomial probability. In this case, we want to find the probability of 6 successes, where a success is making the 3-point shot. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 6, 14, and 0.381, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.197984, which is 0.198 rounded to three decimal places.

On average, Ruth has noticed that 19 trains pass by her house daily (24 hours) on the nearby train tracks. What is the probability that more than 4 trains will pass her house in a 4 hour time period? Round the final answer to three decimal places.

Correct answers:$0.214$0.214​ The time interval of interest is 4 hours. There is an average of 19 trains per 24 hours or 19244=196 trains per 4 hours. The probability can be found using the Poisson distribution with parameter λ=196. Let X be the number of trains that pass in the 4 hour time period. According to the formula, we find: P(X=x)=λxe−λx! Since it is easier to find X equal to or less than 4, we use the compliment:P(X>4)=1−(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4))P(X>4)=1−((196)0e−1960!+(196)1e−1961!+(196)2e−1962!+(196)3e−1963!+(196)4e−1964!)1−(0.042+0.133+0.211+0.223+0.177)≈0.214

Suppose your favorite celebrity posts an average of 2.3 messages per day on a particular social media website. Find the probability that she posts exactly one message for any random day. Round your answer to three decimal places

Correct answers:$0.231$0.231​ Note that this is a Poisson probability. In this case, we want to find the probability of exactly 1 occurrence for a day. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 1 and 2.3, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.230595, which is 0.231 rounded to three decimal places

Suppose your favorite celebrity posts an average of 2.3 messages per day on a particular social media website. Find the probability that she posts exactly one message for any random day. Round your answer to three decimal places.

Correct answers:$0.231$0.231​ Note that this is a Poisson probability. In this case, we want to find the probability of exactly 1 occurrence for a day. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 1 and 2.3, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.230595, which is 0.231 rounded to three decimal places.

On average, Mary has noticed that 18 trains pass by her house daily (24 hours) on the nearby train tracks. What is the probability that exactly 1 train will pass her house in a 3 hour time period? Round the final answer to three decimal places.

Correct answers:$0.237$0.237​ The time interval of interest is 3 hours. There is an average of 18 trains per 24 hours or 18243=94 trains per 3 hours. The probability can be found using the Poisson distribution with parameter λ=94. Let X be the number of trains that pass in the 3 hour time period. According to the formula, we find: P(X=x)=λxe−λx! P(X=1)=(94)1e−941! ≈0.237

Alison is playing a racing video game, in which she can pick up items from item boxes. One of the possible items is a speed booster, which has a 1 in 12 chance of appearing in any item box. What is the probability that Allison will get her first speed booster in one of her first five item boxes? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.353$0.353​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be getting a speed booster from an item box. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most five trials, there are at most four failures before the first success. Thus, enter 4, 1, and 1/12, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.352772, which is 0.353 rounded to three decimal places.

Isaac, a manager at a supermarket, is inspecting cans of pasta to make sure that the cans are not dented and do not have other defects. From past experience, he knows that 1 can in every 80 is defective. What is the probability that Isaac will find his first defective can among the first 45 cans? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.432$0.432​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be finding a defective can. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 45 trials, there are at most 44 failures before the first success. Thus, enter 44, 1, and 1/80, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.432234, which is 0.432 rounded to three decimal places.

A roulette wheel has 38 slots, numbered 1 to 36, with 2 additional green pockets labeled 0 and 00. Jim, a dealer at a roulette table, tests the roulette wheel by spinning a ball around the wheel repeatedly and seeing where the ball lands. What is the probability that Jim will spin the wheel at most 11 times before landing a ball in one of the green pockets? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.448$0.448​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be landing a ball in one of the green pockets. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 11 trials, there are at most 10 failures before the first success. Thus, enter 10, 1, and 2/38, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.448294, which is 0.448 rounded to three decimal places.

Greg, a pharmacist, is reviewing recent prescriptions to determine if they were overfilled, underfilled, contained the wrong medicine, or had other errors. From past experience, he knows that 1 prescription in every 130 had an error. What is the probability that Greg will review at most 80 prescriptions before finding a prescription with an error? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.461$0.461​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be finding a prescription with an error. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 80 trials, there are at most 79 failures before the first success. Thus, enter 79, 1, and 1/130, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.460851, which is 0.461 rounded to three decimal places.

Jackie, a safety consultant, was brought in to a factory to assess the factory's safety procedures. The factory has an average of 0.47 injury per month. Find the probability that there will be no injuries in the factory on a randomly selected month. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.625$0.625​ Note that this is a Poisson probability. In this case, we want to find the probability of 0 occurrences for a day. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 0 and 0.47, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.625002, which is 0.625 rounded to three decimal places.

Give the numerical value of the parameter p in the following binomial distribution scenario.A softball pitcher has a 0.721 probability of throwing a strike for each pitch and a 0.279 probability of throwing a ball. If the softball pitcher throws 19 pitches, we want to know the probability that more than 15 of them are strikes.Consider strikes as successes in the binomial distribution. Do not include p= in your answer.

Correct answers:$0.721$0.721​ The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a strike, so p=0.721.

In a recent baseball season, Bob hit a home run approximately once every 18.38 plate appearances. Assume that this probability did not change going into the next season. What is the probability that Bob hits his first home run before his 25th plate appearance of the season? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.739$0.739​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be hitting a home run. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 24 trials, there are at most 23 failures before the first success. Thus, enter 23, 1, and 1/18.38, in that order. In the entry for a cumulative probability, enter 1 since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.738843, which is 0.739 rounded to three decimal places.

A roulette wheel has 38 slots, numbered 1 to 36, with two additional green slots labeled 0 and 00. Jim, a dealer at a roulette table, tests the roulette wheel by spinning a ball around the wheel repeatedly and seeing where the ball lands. The ball has an equally likely chance of landing in each slot. If Jim spins the ball around the wheel 25 times, what is the probability that the ball lands in a green slot at most twice? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.858$0.858​ Note that this is a binomial probability. In this case, we want to find the probability of 0 to 2 successes, inclusive, where a success is the ball landing in 0 or 00. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 2, 25, and 2/38, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.857891, which is 0.858 rounded to three decimal places

A steel mill manufactures steel plates. The average number of flaws in the steel is 1.28 flaws per square foot. Find the probability that a randomly selected 1-foot-by-1-foot area of steel plate has 2 flaws or fewer. Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.862$0.862​ Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 0 to 2 occurrences for a 1-square-foot area, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 2 and 1.28, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.861693, which is 0.862 rounded to three decimal places.

Kevin works for a company that manufactures solar panels. In a large batch of solar panels, about 1 in 200 is defective. Suppose that Kevin selects a random sample of six solar panels from this batch. What is the probability that none of the solar panels are defective? Use Excel to find the probability. Round your answer to three decimal places.

Correct answers:$0.970$0.970​ Note that this is a binomial probability. In this case, we want to find the probability of 0 successes, where a success is a defective solar panel. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 0, 6, and 1/200, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.970373, which is 0.970 rounded to three decimal places.

A bakery gets an average of 20 phone orders in a 2 hour time period. In order to find the probability that the bakery will get exactly 2 phone orders in a 15 minute period using the Poisson distribution, what is the time interval of interest?

Correct answers:$15\ \min$15 min​ The time interval of interest is the fixed time period for which the probability of an event is being sought. In this case, the time interval of interest is 15 minutes.

Give the numerical value of the parameter n in the following binomial distribution scenario.The probability of buying a movie ticket with a popcorn coupon is 0.597 and without a popcorn coupon is 0.403. If you buy 18 movie tickets, we want to know the probability that no more than 13 of the tickets have popcorn coupons.Consider tickets with popcorn coupons as successes in the binomial distribution. Do not include n= in your answer.

Correct answers:$18$18​ The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, the total number of trials, or movie tickets, is n=18.

The probability of buying a box of cereal with a winning coupon is 0.171. If you buy boxes of this cereal until you get the winning coupon for the first time, what is the expected number of purchases it will take? Round your answer to the nearest whole number.

Correct answers:$6$6​ This is an example of a geometric distribution with success probability p=0.171. The expected value of the random variable of a geometric distribution is 1p=10.171=5.85. Therefore, the expected number of purchases is 6.

A weighted die has probability 0.176 of landing on a 6. If you roll the die until you get a 6 for the first time, what is the expected number of rolls it will take?

Correct answers:$6\text{ rolls}$6 rolls​ This is an example of a geometric distribution with success probability p=0.176. The expected value of the random variable of a geometric distribution is 1p=10.176≈5.68. Therefore, the expected number of rolls is 6.

A guessing game at a casino features 50 cards labeled with the numbers 1 through 50. Four cards will be drawn without replacement and each player will guess the card numbers. The probability of each payout amount is shown in the table (assume that the remaining probability has a payout of 0 so that the probabilities add to 1). To the nearest dollar, what is the expected payout of the game?

Correct answers:$79$79​ The table shows the probability density function where the random variable takes on the values 100, 1400, and 180000. To find the expected value, multiply each payout amount by its probability and round to the nearest dollar: $100(0.106)+$1400(0.023)+$180000(0.0002)=$79.

The table below shows an incomplete probability density function for the discrete random variable X, the number of candy bars purchased per customer at a movie theater. What should the missing probability be? Provide the final answer as a fraction.

Correct answers:$\frac{1}{4}$14​​ Remember that the probabilities in the probability density function must add up to 1. So if we let the unknown value be A, we find that A+14+12=1 So solving for A, we find that A=14.

A restaurant gets an average of 12 phone orders in a 2 hour time period. In order to find the probability that the restaurant will get exactly 2 phone orders in a 30 minute period using the Poisson distribution, what is the average number of phone orders per 30 minutes?

Correct answers:$\lambda=3$λ=3​ The ratio of the average number of phone orders in 2 hours is equal to the ratio of the average number of phone orders in 30 minutes. Let λ represent the average number of phone orders in 30 minutes, then λ30=12120. Solving this equation for λ shows that λ=12(30)120=3.

Christine has a motion detector light which gets activated an average of 16 times every 2 hours during the night. In order to find the probability that the motion detector light will be activated more than 4 times in a 25 minute period during the night using the Poisson distribution, what is the average number of activations per 25 minutes? Round your answer to three decimal places.

Correct answers:$\lambda=3.333$λ=3.333​ The ratio of the average number of activations in 2 hours is equal to the ratio of the average number of activations in 25 minutes. 2 hours is equivalent to 120 minutes. Let λ represent the average number of activations in 25 minutes, then λ25=16120. Solving this equation for λ shows that λ=16(25)120=3.333.

Elizabeth gets an average of 10 emails during her 8 hour work day. In order to find the probability that Elizabeth will get more than 3 emails in a 3 hour portion of her work day using the Poisson distribution, what is the average number of emails received per 3 hours? Round your answer to two decimal places.

Correct answers:$\lambda=3.75$λ=3.75​ The ratio of the average number of emails in 8 hours is equal to the ratio of the average number of emails in 3 hours. Let λ represent the average number of emails in 3 hours, then λ3=108. Solving this equation for λ shows that λ=10(3)8=3.75.

Connor is a statistics student interested in the number of games won by each team per season during the past 12 years for a certain professional baseball league. He records the total number of wins x for each team each season and the probability of each value P(x), as shown in the table provided. Use Excel to find the mean and the standard deviation of the probability distribution. Round the mean and standard deviation to three decimal places.

Correct answers:$\mu=69.928,\ \sigma=6.965$μ=69.928, σ=6.965​ Step 1: Enter the values of the random variable X in column A and the corresponding probabilities in column B. Then find the product of column A and column B and enter the results in column C. Step 2: Add the elements in column C to find the mean μ, which is 69.928. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 4,938.434. Step 4: Calculate the variance by subtracting ∑x2P(x) and the square of the mean. The variance is approximately 48.509. Step 5: Find the square root of the variance to find the standard deviation. The standard deviation, rounded to three decimal places, is σ≈6.965.

An insurance office records the number of claims received each day, X, and built the probability distribution table below using the data collected. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to two decimal places.

Correct answers:$\text{mean}=4.73\text{, standard deviation}=4.67$mean=4.73, standard deviation=4.67​ The mean and the standard deviation of the probability distribution can be calculated using Excel. Step 1: Place the values of the random variable, x, in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Add the elements in the third column to find the mean, μ, rounded to two decimal places, which is 4.73. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 44.113. Step 4: Calculate the variance by subtracting the square of the mean from ∑x2P(x). The variance, rounded to four decimal places, is 21.7817. Step 5: Find the square root of the variance to find the standard deviation. The standard deviation, σ, rounded to two decimal places, is 4.67.

A survey asked recently retired people the number of people, X, they met through work during their working lives that they considered close friends. The table below is a probability distribution table representing the data collected. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to three decimal places.

Correct answers:$\text{mean}=8.783\text{, standard deviation}=5.333$mean=8.783, standard deviation=5.333​ The mean and the standard deviation of the probability distribution can be calculated using Excel. Step 1: Place the values of the random variable, x, in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Add the elements in the third column to find the mean, μ, which is 8.783. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 105.579. Step 4: Calculate the variance by subtracting the square of the mean from ∑x2P(x). The variance, rounded to three decimal places, is 28.438. Step 5: Find the square root of the variance to find the standard deviation. The standard deviation, σ, rounded to three decimal places, is 5.333.

The table below represents the probability density function for the random variable X. Find the standard deviation of X. Round the final answer to two decimal places. x - 0, 2, 6 P(X=x) - 1/5, 2/5, 2/5

Correct answers:$\text{std=}2.40$std=2.40​ Here's the filled out table. x026P(X=x)152525x⋅P(X=x)045125(x−μ)2P(X=x)2.0480.5763.136 Note that the mean (also called expected value) μ is the sum of the entries in the third column:μ=45+125=3.2This value is used to compute the values in the fourth column. For example, the first entry is(0−3.2)2⋅15=2.048Summing the values in the fourth column gives the variance:2.048+0.576+3.136=5.76Taking the square root of the variance gives the standard deviation:σ=5.76−−−−√≈2.40

Barbara is a quality control inspector at a shoe factory. At the end of each day, she checks the number of imperfections found in sneakers. The table below represents the probability density function for the random variable X, the number of imperfections found in sneakers per day. Find the standard deviation of X. Round the final answer to two decimal places. x P(X = x) 0 1/4 2 1/4 4 1/4 7 1/4

Correct answers:$\text{std=}2.59\text{ imperfections}$std=2.59 imperfections​ Here's the filled out table. x0247P(X=x)14141414x⋅P(X=x)012174(x−μ)2P(X=x)2.64060.39060.14063.5156 Note that the mean (also called expected value) μ is the sum of the entries in the third column:μ=12+1+74=3.25This value is used to compute the values in the fourth column. For example, the first entry is(0−3.25)2⋅14=2.6406Summing the values in the fourth column gives the variance:2.6406+0.3906+0.1406+3.5156=6.6874Taking the square root of the variance gives the standard deviation:σ=6.6874−−−−−√≈2.59

Andrew is a quality control inspector at a clothing factory. At the end of each day, he checks the number of imperfections found in cotton sweaters. The table below represents the probability density function for the random variable X, the number of imperfections found in cotton sweaters per day. Find the standard deviation of X. Round the final answer to two decimal places. x P(X = x) 0 1/6 1 1/6 4 1/3 7 1/3

Correct answers:$\text{std=}2.67\text{ imperfections}$std=2.67 imperfections​ Here's the filled out table. x0147P(X=x)16161313x⋅P(X=x)0164373(x−μ)2P(X=x)2.44911.3380.00933.3426 Note that the mean (also called expected value) μ is the sum of the entries in the third column:μ=16+43+73=3.8333This value is used to compute the values in the fourth column. For example, the first entry is(0−3.8333)2⋅16=2.4491Summing the values in the fourth column gives the variance:2.4491+1.338+0.0093+3.3426=7.139Taking the square root of the variance gives the standard deviation:σ=7.139−−−−√≈2.67

During a college football national championship, the number of times football teams scored a touchdown was recorded by a scorekeeper every quarter. The table below represents the probability density function for the random variable X, the number of touchdowns per quarter. Find the standard deviation of X. Round the final answer to two decimal places. x P(X = x) 1 1/5 2 1/5 8 1/5 9 2/5

Correct answers:$\text{std=}3.54\text{ touchdowns}$std=3.54 touchdowns​ Here's the filled out table. x1289P(X=x)15151525x⋅P(X=x)152585185(x−μ)2P(X=x)4.6082.8880.9684.096 Note that the mean (also called expected value) μ is the sum of the entries in the third column:μ=15+25+85+185=5.8This value is used to compute the values in the fourth column. For example, the first entry is(1−5.8)2⋅15=4.608Summing the values in the fourth column gives the variance:4.608+2.888+0.968+4.096=12.56Taking the square root of the variance gives the standard deviation:σ=12.56−−−−√≈3.54

The probability of winning on an arcade game is 0.659. If you play the arcade game 30 times, what is the probability of winning exactly 21 times? Round your answer to two decimal places.

Correct answers:1$0.14$0.14​ In this case, we have that x=21, n=30, and p=0.659. Thus, the probability can be found by, P(X=21)=30C21⋅0.65921⋅0.3419=30!21!9!⋅0.65921⋅0.3419≈0.14

A study randomly selected 100 samples, each of which consisted of 100 people, and recorded the number of left-handed people, X. The table below shows the probability distribution of the data. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to two decimal places.

Correct answers:= mean = 11.56, standard deviation=4.85​ The mean and the standard deviation of the probability distribution can be calculated using Excel. Step 1: Place the values of the random variable, x, in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Add the elements in the third column to find the mean, μ, which is 11.56. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 157.2. Step 4: Calculate the variance by subtracting the square of the mean from ∑x2P(x). The variance is 23.5664. Step 5: Find the square root of the variance to find the standard deviation. The standard deviation, σ, rounded to two decimal places, is 4.85

Discrete random variable: variables whose possible values are a list of distinct values Mean: the sum of all the items in a list divided by the number of items in the listThe Mean is sometimes referred to as the Average Standard deviation: a number that measures how far data values or probability distributions are from their mean, denoted by a Greek letter (σ)

Discrete random variable: variables whose possible values are a list of distinct values Mean: the sum of all the items in a list divided by the number of items in the listThe Mean is sometimes referred to as the Average Standard deviation: a number that measures how far data values or probability distributions are from their mean, denoted by a Greek letter (σ)

Calculate Mean and Standard Deviation for a Discrete Probability Distribution

Find the mean and standard deviation for random variable X using the table given below. Probability Distribution Table x= 3, 4, 5 P(x) = 0.25, 0.5, 0.25 Step 1: Write the values of the random variable X in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Drag the bottom right corner of cell C2 to the end of the table. Add the elements in the third column to find the mean μ Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B (for example, the formula for cell D2 would be A2*A2*B2). Drag the bottom right corner of cell D2 to the end of the table. Now add the elements in the fourth column to find ∑x2P(x). Step 4: Calculate the variance by subtracting ∑x2P(x) and the square of the mean Step 5: Find the square root of the variance to find the standard deviation.

The outcomes of a binomial experiment fit a binomial probability distribution

Here is the notation for a binomial distribution of a random variable X: X∼B(n,p). Read this as "X is a random variable with a binomial distribution." The parameters inside the parentheses are n and p, which are the values of the number of trials n and the probability of a success on each trial p

A softball pitcher has a 0.431 probability of throwing a strike for each pitch. If the softball pitcher throws 22 pitches, what is the probability that exactly 12 of them are strikes? Round your answer to 2 decimal places.

In this case, we have that x=12, n=22, and p=0.431. Thus, the probability can be found by,B(12;22,0.431)=22C12⋅0.43112⋅0.56910=22!12!10!⋅0.43112⋅0.56910≈646646⋅0.00004⋅0.00356≈0.09

In a 40-year period, a city's official weather station recorded 78 snowstorms with at least 12 inches of total snow accumulation. Assume that this pattern follows a Poisson distribution. Find the probability that the city's weather station records exactly 4 snowstorms with at least 12 inches of total snow accumulation for one randomly selected year in the period. Use Excel to find the probability. Round your answer to three decimal places.

Note that this is a Poisson probability. In this case, we want to find the probability of 4 occurrences for a year. The mean for the Poisson distribution is 7840=1.95 occurrences per year. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 4 and 1.95, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.085714, which is 0.086 rounded to three decimal places.

A state lottery sells instant-lottery scratch tickets. 12% of the tickets have prizes. Neil goes to the store and buys 10 tickets. What is the probability that exactly three of Neil's tickets will have prizes? Use Excel to find the probability. Round your answer to three decimal places

Note that this is a binomial probability. In this case, we want to find the probability of 3 successes, where a success is a ticket having a prize. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 3, 10, and 0.12, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.084743, which is 0.085 rounded to three decimal places.

An animal rescue organization computes that there are an average of 8.2 feral cats per acre of uninhabited land in a certain city. Find the probability that there are fewer than 6 feral cats living on a random acre of uninhabited land in the city.

Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 0 to 5 occurrences, inclusive. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 5 and 8.2, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.173594.

A baseball park ran a "$1 Hot Dog Night" promotion. The average number of hot dogs purchased that night was 3.3 hot dogs per fan. Find the probability that a randomly selected fan who attended $1 Hot Dog Night purchased 5 or more hot dogs. Round your answer to 3 decimal places.

Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 5 or more occurrences for a fan. This is the complement of the probability of 0 to 4 occurrences for a fan, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 4 and 3.3, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.76259, which is 0.763 rounded to three decimal places. To find the probability of the fan buying 5 or more hot dogs, subtract this probability from 1. The probability is 1−0.763=0.237

At a university, 64% of the student body lives on campus. Find the probability that at most 12 students in a class of 25 live on campus, where this class is considered to be a random sample of all students.

Note that this is a cumulative binomial probability. In this case, we want to find the probability of 0 to 12 successes, inclusive, where a success is a student that lives on campus. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. Select a blank cell. Press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 12, 25, and 0.64, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.074493.

The probability of buying a movie ticket with a popcorn coupon is 0.717. If you buy 21 movie tickets, what is the probability that exactly 14 of the tickets have popcorn coupons? (Round your answer to 3 decimal places if necessary.)

P(X=14)=1$$ Correct answers:1$0.160$0.160​ In this case, we have that x=14, n=21, and p=0.717. Thus, the probability can be found by, P(X=14)=21C14⋅0.71714⋅0.2837=21!14!7!⋅0.71714⋅0.2837≈0.160

Poisson Distribution:

Poisson Distribution: a discrete probability distribution that gives the probability of a number of events occurring in a fixed interval of time or space

Random variable: a variable that is subject to change due to chance in events and describes the outcomes of a statistical experiment Sample space: all possible outcomes or results for an experiment Discrete data: data that you can count Discrete random variable: variables whose possible values are a list of distinct values

Random variable: a variable that is subject to change due to chance in events and describes the outcomes of a statistical experiment Sample space: all possible outcomes or results for an experiment Discrete data: data that you can count Discrete random variable: variables whose possible values are a list of distinct values

A casino features a game in which a weighted coin is tossed several times. The table shows the probability of each payout amount (assume that the remaining probability has a payout of 0 so that the probabilities add to 1). To the nearest dollar, what is expected payout of the game?

The table shows the probability density function where the random variable takes on the values 130, 2800, and 135000. To find the expected value, multiply each payout amount by its probability and round to the nearest dollar: $130(0.131)+$2800(0.021)+$135000(0.0003)≈$116.

A gambling game involves a spinner with the numbers 1 through 25. To play, the player guesses which numbers the spinner will land on. The table shows the probability of each payout amount (assume that the remaining probability has a payout of 0 so that the probabilities add to 1). To the nearest dollar, what is the expected payout of the game?

The table shows the probability density function where the random variable takes on the values 140, 4400, and 170000. To find the expected value, multiply each payout amount by its probability and round to the nearest dollar: $140(0.116)+$4400(0.021)+$170000(0.0007)=$228.

On average, Benjamin has noticed that 20 trains pass by his house daily (24 hours) on the nearby train tracks. What is the probability that exactly 9 trains will pass his house in a 12 hour time period? Round your answer to three decimal places.

The time interval of interest is 12 hours. There is an average of 20 trains per 24 hours or 20//24/12=10 trains per 12 hours. The probability can be found using the Poisson distribution with parameter λ=10. Let X be the number of trains that pass in the 12 hour time period. According to the formula, we find: P(X=x)=λxe−λ/x! P(X=9)=109e−109! ≈0.125

binomial distribution.

There are three characteristics of a binomial experiment. 1. There are a fixed number of trials, or repetitions of an experiment. The letter n denotes the number of trials. 2. There are only two possible outcomes, called "success" and "failure," for each trial. The letter p= the probability of a success of a trial. The letter q= the probability of a failure on one trial. So, p+q=1. 3. The n trials are independent and are repeated using identical conditions. Because they are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same For example, the probability that Joe randomly guesses the correct answer on a test is 0.6. We will say that guessing correctly is a success, so p=0.6. Since there are only two outcomes for this event - guessing correctly, or guessing incorrectly - this is a binomial distribution. So, then a failure is guessing incorrectly. Since p+q=1, and p=0.6, we can determine Joe's probability of failure too, q=0.4 (because 0.6+0.4=1).

Find a Poisson Probability with Excel

To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 5 and 10, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.037833. Note: The last entry would be a 1 to find a cumulative probability.

An urn contains 50 total balls, which comprise 49 white balls and one green ball. Dwight is running an experiment where he draws a ball from the urn, records the color, and replaces the ball, repeating until he draws the green ball. What is the probability that Dwight will take at most 10 tries to draw the green ball from the urn? Use Excel to find the probability, rounding to three decimal places

Use Excel to find the probability. Note that this is a cumulative probability. Let a success be drawing the green ball. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 10 trials, there are at most 9 failures before the first success. Thus, enter 9, 1, and 1/50, in that order. In the entry for a cumulative probability, enter 1 since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.182927, which is 0.183 rounded to three decimal places.

An urn contains 4 total balls, which comprise 3 white balls and one green ball. Dwight is running an experiment where he draws a ball from the urn, records the color, and replaces the ball, repeating until he draws the green ball. What is the probability that Dwight will take exactly 4 tries to draw the green ball from the urn? Use Excel to find the probability, rounding to three decimal places.

Use Excel to find the probability. Note that this is not a cumulative probability. Let a success be drawing the green ball. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of exactly 4 trials, there are 3 failures before the first success. Thus, enter 3, 1, and 1/4, in that order. In the entry for a cumulative probability, enter 0 since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.105.

Poisson Distribution

a discrete probability distribution that gives the probability of a number of events occurring in a fixed interval of time or space

random variable

is a variable that is subject to change due to chance in events. It describes the outcomes of a statistical experiment, and it can take on a set of different possible values. Each random variable value can vary with each repetition of an experiment, and each one has an associated probability (in contrast to other mathematical variable Notation for a random variable is an upper case letter, usually X or Y. These variables are usually represented with words, and do not have numerical values. (i.e. "The random variable X represents the number of blue cars on the highway.") Lower case letters like x or y denote the value of a random variable, and x is given as the numerical value of the random variable.

Rae packs an average of 45 boxes during her 3 hour shift. In order to find the probability that Rae will pack at least 55 boxes in a 4 hour shift using the Poisson distribution, what does the random variable X represent?

the number of boxes she packs in 4 hours The random variable X represents the number of occurrences of an event in the time interval of interest. The time interval of interest is the fixed time period for which the probability of an event is being sought. In this case, the time interval of interest is 4 hours. The random variable X is the number of boxes that Rae packs in that amount of time.