Stats TBL 8
Consider the sample space of the random experiment rolling a fair die: S=(1,2,3,4,5,6). Consider event A=(1,3,5), then the compliment of A contains...
(2, 4, 6)
Consider two mutually exclusive events, A and B. The probability of A is 0.7, and the probability of B is 0.2. What is the probability of A intersect B?
0 If two events are mutually exclusive, then they have no intersection. If no intersection exists, then the probability of A intersect B is zero.
Apply Bayes? Theorem to calculate P(B|A) with the following information: P(A|B)=0.9; P(A|B?)=0.1; P(B)=0.2, where an apostrophe represents the complement of the event. Round to the closest value.
0.69 Pic
What is the probability of getting between 4 and 7 heads (inclusive) in 10 tosses of a fair coin?
0.773 A Binomial experiment occurs when a Bernoulli experiment is repeated n times where each time is independent of the other trials. Recall that a Bernoulli experiment is a random experiment that has only two possible outcomes (success or failure).An example of a Bernoulli experiment is flipping a coin. Here only two outcomes exist, heads or tails. If this coin flip is repeated, let's say three times, then it now has a Binomial distribution.The binomial probability formula is as follows: n - Number of Trials = 10 π - Probability of Success = 0.5 x1 - Number of Success = 4 x2 - Number of Success = 7 ! - Factorial (i.e. 5! = 5 x 4 x 3 x 2 x 1) P(4 ≤ X ≤ 7) = P(X=4) + P(X=5) + P(X=6) + P(X=7) P(X=5) = 0.2461 P(X=6) = 0.2051 P(X=7) = 0.1172 P(4 ≤ X ≤ 7) = 0.2051 + 0.2461 + 0.2051 + 0.1172 = 0.7735
Consider two independent events, A and B. P(A) = 0.6, P(B) = 0.5. Then P(A or B) =
0.8 0.6 + 0.5 - 0.6 * 0.5 1.1 - 0.30 0.8
What is the expected value of the following random variable? X. P(X) 0. 0.3 1 0.5 2. 0.2
0.90 Multiply across then add
Consider the sample space of the random experiment of rolling a fair die S=(1,2,3,4,5,6). Consider the two events: A=(1,3,6), and B=(2,4,5). Then P(A and B) =...
0/6 6 - 3 -3
The probabilities on the branches immediately following the initial node must sum to
1
How many ways can 8 employees be selected from a group of 20 applicants?
125,970 (Combinations calculator)
Consider the sample space of the random experiment rolling a fair die: S=(1,2,3,4,5,6). Consider event A=(1,4,5), then P(A)=...
3/6 6-3
Consider the random experiment to tossing two fair coins. What is the total number of outcomes of the random experiment?
4
Consider the sample space of the random experiment of rolling a fair die S=(1,2,3,4,5,6). Consider the two events: A=(1,3), and B=(3,4,6). Then P(A or B) =...
4/6
If you flip a coin three times, the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. What is the probability of getting at least one head?
7/8 You can use classical probability and the complement rule to calculate the probability, since the probability of at least 1 head is equal to 1 - P(No heads). (The complement of at least one head is No heads.)
Which of the following experiments would have a sample space that is continuous
Asking 10 students how long their commute to class is Time is a continuous variable, whereas how many coin tosses came up heads or how many pairs of shoes an individual owns are discrete.
When the outcome of each trial can assume only two possibilities, and the trials are Statistically Independent, then the random variable that tracks the number of successes in those trials will follow which distribution?
Binomial
Consider the random experiment of observing the gender of costumers entering the 7-11 convenience store. What type of sample space is associated with this experiment?
Countably Finite
How would you find the marginal probability of an event from a contingency table?
Divide the row or column total by the total sample size.
Suppose you have data on the number of times Zach is late for class. You compute the probability that he will be late for the next class to computing the relative frequency of times he is late out of the number of sessions he attended. The approach to assigning probabilities is the (i) approach
Emperical
You rolled a die 10 times, and 5 times the 6 side was face up so you decided there is a 50% chance that if you roll a die you get a 6. Which approach did you use?
Empirical Empirical Approach: Collecting data through observations or experiments. For example, flipping a coin ten times and concluding that there is a 60% chance of getting heads because heads came up 6 out of the 10 times you flipped the coin. Classical Approach: Assigning probabilities before actually observing the event or trying an experiment. For example, it makes sense that, given a fair coin, there would be an equal chance of getting heads or tails. Subjective Approach: This is someone's informed judgement about the likelihood of an event. This can be useful when there is not an experiment that can be performed to figure it out. Assessing the New York Knick's chances of an NBA title next year would be an example.
Consider two events, A and B. The probability of A is 0.4, the probability of B is 0.4, and the probability of A intersect B is 0.2. True or false: these two events are independent
False
True or False: If P(A and B) = 1, then A and B must be mutually exclusive
False
Two events are considered to be mutually exclusive if the events are also independent.
False
Two events are considered to be mutually exclusive if the events are also independent
False Recall the definition of mutually exclusive events. If events are mutually exclusive, then the joint event can not occur. Now, look at Probability Rule 8. We can only conclude that events are independent if the joint event occurs. In fact, if the occurrence of event A precludes the occurrence of event B, then we would certainly conclude they are dependent events!
When a quarter is tossed four times, 16 outcomes are possible: HHHH, HHHT, HHTH, HHTT HTHH, HTHT, HTTH, HTTT THHH, THHT, THTH, THTT TTHH, TTHT, TTTH, TTTT Here, for example, HTTH represents the outcome that the first toss is heads, the next two tosses are tails, and the fourth toss is heads.The event A is defined as follows. A = event the first two tosses are headsList all the outcomes that comprise the event (not A)
HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT The table displaying the sample space is conveniently arranged so that only those outcomes in the first row satisfy event A. Thus, the remaining 12 outcomes are the ones that satisfy the complement, (not A).
List the outcomes comprising the specified event. When a quarter is tossed four times, 16 outcomes are possible: HHHH, HHHT, HHTH, HHTT HTHH, HTHT, HTTH, HTTT THHH, THHT, THTH, THTT TTHH, TTHT, TTTH, TTTT Here, for example, HTTH represents the outcome that the first toss is heads, the next two tosses are tails, and the fourth toss is heads. List the outcomes that comprise the following event. A = event exactly three tails are tossed
HTTT, THTT, TTHT, TTTH A tree diagram of what can happen on each of the 4 tosses will help with this one. Or, just focus on the number of ways you can get exactly 3 tails in 4 tosses. There must be one head in each quadruple, just rotate the H to a different position until you found all the possibilties.
In a classroom there are 106 students. 55 of them are male. There are 42 seniors in the class and 16 of the females are seniors. What is the probability that a randomly chosen student is a male or a senior?
Here we utilize the General Law of Addition: Step 1 - Define and Calculate the probabilities P(A∪B) - Probability of A or B - Probability that a randomly chosen student is a male or a senior P(A) - Probability of A - Probability that a randomly chosen student is a male = 55/106 = 0.519 P(B) - Probability of B - Probability that a randomly chosen student is a senior = 42/106 = 0.396 P(A∩B) - Probability of A and B - Probability that a randomly chosen student is a male and a senior = (42-16)/106 = 0.245 Step 2 - Plug the calculated probabilities into the General Law of Addition equation.0.670 = 0.519 + 0.396 - 0.245
Consider two events A and B. The probability of A is 0.4, the probability of B is 0.5, and the probability of A intersect B is 0.1. What is the probability of A union B?
Here you use the General Law of Addition: P(A∪B) - Probability of A or B P(A) - Probability of A = 0.4 P(B) - Probability of B = 0.5 P(A∩B) - Probability of A and B = 0.1 P(A∪B) = P(A) + P(B) - P(A∩B) P(A∪B) = 0.4 + 0.5 - 0.1 P(A∪B) = 0.8
A survey of UMass students finds that the expected value for the discrete random variable "number of credit cards" is 1.76. Interpret this value
If UMass students were repeatedly sampled, we would expect to find that they held 1.76 cards, on average. Feedback: An expected value is an "average" based on knowledge of all possible values. Expected values are parameters for probability distributions. We interpret expected values as "averages."
A Binomial Experiment is a random experiment which consist of n (i) Bernoulli trials
Independent and Identical
n a competition, two people will be selected from four finalists to receive the first and second prizes. The prize winners will be selected by drawing names from a hat. The names of the four finalists are Jim, George, Helen, and Maggie. The possible outcome(s)can be represented as follows: JG JH JM GJ GH GM HJ HG HM MJ MG MH Here, for example, HM represents the outcome that Helen receives the first prize and Maggie receives the second prize. Which outcome list fully comprises the following event? A = event that George wins second prize
JG, HG, MG Just find all the outcomes where old "G" appears second.
What type of probability is the probability that a randomly selected iPhone is defective and is produced by Company A?
Joint
When making a joint probability table, what type of probabilities are contained in the cells except those cells on the margin of the table
Joint probabilities
A binomial probability distribution with .75 as the probability of a success will have a distribution with a shape best described by
Left skewed Any binomial probability distribution with a success probability (i.e., p) greater than 0.5 is skewed to the left. If p < 0.5, it is skewed to the right. If p=0.5, the distribution is perfectly symmetric. Finally, for any value of p not equal to 0.5, the level of skewness becomes less pronounced as n rises.
The general shape of a binomial random variable that has a probability of success of .85 will be
Left skewed or j shaped
A multiple choice test consists of 11 questions. Each question has 5 possible answers of which only one is correct. A student guesses on every question. Find the expected number of correct answers to this test from just guessing.
Let X be a random variable denoting number of correct guesses given by the studentn. Probability of success = 1/5 Total number of trials = 11 Standard deviation is given by : square root of np(1-p) Square root of 11 * 1/5 * 4/5 =1.3266
A compound event is an event containing (i).
More than one outcome
What are the two properties of the events in the sample space from a Bernoulli Experiment?
Mutually Exclusive and Collectively Exhaustive
n a competition, two people will be selected from four finalists to receive the first and second prizes. The prizewinners will be selected by drawing names from a hat. The names of the four finalists are Jim, George, Helen, and Maggie.The possible outcomes can be represented as follows: JG JH JM GJ GH GM HJ HG HM MJ MG MH Here, for example, JG represents the outcome thatJim receives the first prize and George receives the second prize. List the outcome that fully comprises the following event. A = event that both prize winners are women
None of the Above Okay - first, we're counting on you to get Helen (H) and Maggie (M) are the two women. Next, you should note that there are first and second prizes, so order matters. Now the answer should be clear. GH, HG, JH HM, MH, HG HM, MH, HG, MG HM, GM, JM None of the Above
A binomial probability distribution for 1,000 trials with .65 as the probability of a success will have a distribution with a shape best described by:
Normal The key here is that there is a large number of trials. While a binomial probability distribution with a success probability (i.e., p) more than 0.5 is skewed to the left for a small number of trials, with 1,000 trials, the distribution will look normal. For any value of p not equal to 0.5, the level of skewness becomes less pronounced as n rises. At 1,000 trials, there will be no skewness observed.
The general shape of a binomial random variable that has a probability of success of .5 and a large n will be
Normal or bell shaped
In a classroom there are 106 students. 56 of them are male. There are 41 seniors in the class and 25 of the females are seniors. What is the probability that a randomly chosen student is a male and a senior?
P(Male∩Senior) To find the probability that a randomly chosen student is a male and a senior we need to first calculate how many total students are males and seniors (male seniors). Total amount of Male Seniors = Seniors - Female Seniors = 41 - 25 = 16 Next, to calculate this probability we simply divide the number of Male Seniors by the total number of students in the class. P(Male∩Senior) = 16 / 41 = 0.151
As reported in Trends in Television, the proportion of US households who have at least one VCR is 0.308. If 12 households are selected at random, without replacement, from all US households, what is the (approximate) probability that the number of households having at least one VCR is exactly 6. Be sure to use many decimal places in your calculations (at least 4), but report your answer to three decimal places.
Pic
As reported in Trends in Television, the proportion of US households who have at least one VCR is 0.403. If 14 households are selected at random, without replacement, from all US households, what is the (approximate) probability that the number having at least one VCR is no more than 7 but at least 5.00. Be sure to use many decimal places in your calculations (at least 4), but report your answer to three decimal places.
Pic
Consider two events, A and B. The probability of A is 0.3, the probability of B is 0.2, and the probability of A union B is 0.3. What is the probability of A intersect B?
Pic 0.3 + 0.2 - 0.3= 0.2
a multiple choice test consists of 23 questions. each question has 3 possible answers of which only one is correct. a student guessed on every question. find the expected number of correct answers to this test from just guessing. do your calculations with many decimal places (at least 4), but round your answer to 2 decimal places
Pic 23 * 1/3
The general shape of a binomial random variable that has a probability of success of .25 will be
Right skewed or reverse j shaped
The (i) of a discrete random variable X, is the square root of its variance
Standard Deviation
You are set up on a blind date. Based on past experiences you have never liked your date so you assign the scenario a 5% chance that you will like your date. Which approach did you use
Subjective Empirical Approach: Collecting data through observations or experiments. For example, flipping a coin ten times and concluding that there is a 60% chance of getting heads because heads came up 6 out of the 10 times you flipped the coin. Classical Approach: Assigning probabilities before actually observing the event or trying an experiment. For example, it makes sense that, given a fair coin, there would be an equal chance of getting heads or tails. Subjective Approach: This is someone's informed judgement about the likelihood of an event. This can be useful when there is not an experiment that can be performed to figure it out. Assessing the New York Knick's chances of an NBA title next year would be an example.
Consider the random experiment of rolling a pair of fair dice. What is the probability that one of the dice has the number 5 or less facing up given that the other has at least the number 5 facing up?
The roll of the second dice has no effect on the roll of the first dice, thus the events are Independent. When we have two independent events, the formula for conditional probability is as follows: We can define "A" as the number of possible events that have the number 5 or less facing up on die one.Note: we know that the total number of possible events when rolling one die is 6. Therefore,P(A|B) = P(A) = 5 / 6 = 0.83
The collection of all possible events is called
The sample space
Determine the expected number of credit cards for this discrete probability distribution.
To determine the expected value: First use the frequencies to calculate probabilities for each value of the discrete random variable. To calculate a probability, divide each frequency by the total of the frequencies. For example, the probability that a randomly selected student has 1 credit card is 34/112 = 0.304.Second, multiply each value of the random variable (i.e., each possible number of credit cards) by the matching probability that you calculated in the first step. The result will be seven products. Third, sum the seven products from the second step. The result is the expected number of credit cards.
True or False: Another name for the mean of a probability distribution is its expected value.
True
True or False: If p remains constant in a binomial distribution, an increase in n will cause the mean to increase.
True
The expected value of an unbiased estimator is equal to the parameter whose value is being estimated
True The mean (expected value) of the sample mean is equal to the population mean. Thus, the expected value of an unbiased estimator is equal to the parameter whose value is being estimated. However, if the estimator were to be biased than the sample mean would not be equal to the population mean.
Consider two events, A and B. The probability of A is 0.4, the probability of B is 0.5 and the probability of A intersect B is 0.2. True or false: these two events are independent
True Utilizing the special law of multiplication, we can see that events A and B are independent:.5 * .4 = .2
Consider two mutually exclusive events, A and B. The probability of A is 0.4, and the probability of B is 0.6. True or false: these two events are collectively exhaustive
True Since these two events are mutually exclusive, they cannot occur at the same time. Either event A can occur or event B, but not both. Since their respective probabilities of .4 and .6 sum to 1, the two events are collectively exhaustive.
Define: random variable.
a variable that assigns a numerical value to each outcome of a random experiment or trial
If an event "A" has probability 0.65 of occurring, what is the probability of the complement of event "A"?
he complement of an event is the probability that the event does not occur. If A has probability 0.65 then the complement of A is 1-0.65 = 0.35
Expected value
n (x) p
When customers enter a store there are three outcomes that can occur: buy nothing, buy a small amount, or buy a large amount. In this situation if a customer buys a large amount, he or she cannot also buy a small amount or buy nothing. Given this information, we can conclude that these events constitute:
outcomes. mutually exclusive events. the sample space. elementary events. There are three possible outcomes that are defined so that they encompass all possible outcomes or the sample space. But, these events cannot occur at the same time. If a person buys a large amount, by definition they are distinguished from those that "buy a small amount." The events are mutually exclusive. These are also elementary events because they are the "rudimentary outcomes."
In a binomial distribution:
the probability of success p is stable from trial to trial. The binomial random variable is a special one of the general discrete random variables. It requires that trials have only two outcomes (success or failure) and that they are statistically independent, so that the probability of success remains constant from trial to trial.
Simple event: A single outcome Compound event: Two or more simple events Mutually Exclusive events: If two events are mutually exclusive, then only one of them can occur. Independent events: If one event does not affect the probability that another event will occur, then the two events are independent.
x