chapter 6/7
1313 outcomes in A (i.e., 1313 hearts in a deck of cards) and 5252 outcomes in the sample space (i.e., 5252 cards in the deck) the probability of event A is as follows.
P(A) 13/52= 1/4
Consider the event A={annual income is greater than $50,000}A=annual income is greater than $50,000. Suppose the probability of A was 0.08. Determine the probability of observing someone whose income was less than or equal to $50,000
P(annual income less than or equal to 50,000) 1-P(A)= 1-0.08=0.92
The selling prices of 131 randomly selected homes.
discrete
Determine is 33/13 can be a probability
No
Suppose that you and a friend are playing cards and you decide to make a friendly wager. The bet is that you will draw two cards without replacement from a standard deck. If both cards are clubs, your friend will pay you $9. Otherwise, you have to pay your friend $3. Step 1 of 2 : What is the expected value of your bet? Round your answer to two decimal places. Losses must be expressed as negative values.
P(both cards are clubs) = (13/52) * (12/51) = 1/17 a) Expected value of the bet = 9 * (1/17) - 3 * (1 - 1/17) = -$2.29
continuous random variable
a random variable that may assume any numerical value in an interval or collection of intervals
The highway miles per gallon for an automobile
continuous
discrete uniform distribution
is one of the simplest probability distributions. Each value of the random variable is assigned identical probabilities. There are many situations in which the discrete uniform distribution arises
which plan has the least amount of risk Plan A Payout P( Payout ) 35000 0.11 60000 0.5 80000 0.39 Plan B Payout P( Payout ) 15000 0.06 20000 0.37 45000 0.57
plan B
Write out the sample space for the given experiment. Use the following letters to indicate each choice: O for olives, A for asparagus, E for eggs, T for turkey, H for honey mustard, and I for Italian. When deciding what you want to put into a salad for dinner at a restaurant, you will choose one of the following extra toppings: olives, asparagus. Also, you will add one of following meats: eggs, turkey. Lastly, you will decide on one of the following dressings: honey mustard, Italian.
OEH,OEI,OTH,OTI,AEH,AEI,ATH,ATI
Write out the sample space for the given experiment. Use the letter H to indicate heads and T for tails. 3 coins are tossed.
HHH,HHT,HTH,HTT,THH,THT,TTH,TTT
The weights of 89 newborn babies at a local hospital.
continuous
Cars enter a car wash at a mean rate of 4 cars per half an hour. What is the probability that, in any hour, exactly 2 cars will enter the car wash? Round your answer to four decimal places.
0.0107
six sided dice whats the probability of rolling a sum less than or equal to 7?
0.5833
You order some phone covers online and get an estimated delivery date of June 1-June 9. You know you will be out of town June 4th, 5th, and 6th and are a little concerned about the package arriving when you are away. Assuming the delivery date follows a discrete uniform distribution, what is the likelihood your package will be delivered while you are out of town? Round your answer to four decimal places, if necessary.
1/3
A doctor visits her patients during morning rounds. In how many ways can the doctor visit 5 patients during the morning rounds?
5x4x3x2x1=120 ways
Lost-time accidents occur in a company at a mean rate of 0.6 per day. What is the probability that the number of lost-time accidents occurring over a period of 10 days will be no more than 3? Round your answer to four decimal places.
= 0.6 * 10 = 6 P(X = x) = e/x! P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = e^(-6) * 6^0/0! + e^(-6) * 6^1/1! + e^(-6) * 6^2/2! + e^(-6) * 6^3/3! = 0.1512
permutation
An arrangement, or listing, of objects in which order is important.
Cars enter a car wash at a mean rate of 3 cars per half an hour. What is the probability that, in any hour, exactly 5 cars will enter the car wash? Round your answer to four decimal places.
Average = 3 per half an hour = 6 per hour So, P(X = 5) = 0.1606
How many ways can a person toss a coin 10 times so that the number of heads is between 6 and 8 inclusive?
By the Combination Rule, the number of ways a person can toss a coin 10 times and get r heads is 10Cr. In this problem, r is permitted to range between the values of 6 and 8. Consequently, the solution to the problem is obtained by computing 10Cr for each one of these possible values of r and then taking the sum of these computations. 10c6=210 10c7=120 10c8=45 375
Cars enter a car wash at a mean rate of 2 cars per half an hour. What is the probability that, in any hour, exactly 2 cars will enter the car wash? Round your answer to four decimal places.
Here, λ = 2*2 4 and x = 2As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x! We need to calculate P(X = 2)P(X = 2) = 4^2 * e^-4/2!P(X = 2) = 0.1465Ans: 0.1465
Choosing a student who is a communications major or a philosophy major. Mutually exclusive?
Mutually exclusive
Heart or spade out of a standard deck of cards
Mutually inclusive
Red card or spade out of a standard deck of cards
Mutually inclusive
-0.04 be a probability
No
The number of calls received by an office on Monday morning between 8:00 AM and 9:00 AM has a mean of 6. Calculate the probability of getting no more than 2 calls between eight and nine in the morning. Round your answer to four decimal places
Now the probability of getting no more than 2 calls between eight and nine in the morning is computed here as: 0.0620
The number of calls received by an office on Monday morning between 8:00 AM and 9:00 AM has a mean of 2. Calculate the probability of getting at least 4 calls between eight and nine in the morning. Round your answer to four decimal places.
P(X=0)=e−2(2)0/0!=0.1353P(X=0)=e−2(2)00!=0.1353 P(X=1)=e−2(2)1/1!=0.2707P(X=1)=e−2(2)11!=0.2707 P(X=2)=e−2(2)2/2!=0.2707P(X=2)=e−2(2)22!=0.2707 P(X=3)=e−2(2)3/3!=0.1804 0.1429
A standard pair of six-sided dice is rolled. What is the probability of rolling a sum of 2?
S=36 the number of combinations that have sums of 2= 1 probability of rolling a 2= 0.028
odds against
The odds against an event A occurring is given by P(notA)/P(A)=P(Ac)/P(A).
odds in favor
The odds in favor of an event A occurring is given by P(A)/P(notA)=P(A)/P(Ac)
outcome set
The set of all possible outcomes for a given experiment
discrete random variable
Variable where the number of outcomes can be counted and each outcome has a measurable and positive probability
Can 0.94 be a probability
Yes
random variable
a variable whose value is a numerical outcome of a random phenomenon
Birth years of members of your family.
discrete
The number of people who quit smoking in each of the last ten years.
discrete
The number of pizzas delivered to a college campus each day
discrete
Fundamental Counting Principle
if an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in N x M ways. E1 is an event with n1 possible outcomes and E2 is an event with n2 possible outcomes. The number of ways the events can occur in sequence is n1⋅n2. This principle can be applied for any number of events occurring in sequence.
binomial experiment
is a random experiment which satisfies all of the following conditions. There are only two outcomes in each trial of the experiment. (One of the outcomes is usually referred to as a success, and the other as a failure.) The experiment consists of n identical trials as described in Condition 11. The probability of success on any one trial is denoted by p and does not change from trial to trial. (Note that the probability of a failure is (1−p)1−pand also does not change from trial to trial.) The trials are independent. The binomial random variable is the count of the number of successes in ntrials.
At the Fidelity Credit Union, a mean of 5.5 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 5 customers will arrive? Round your answer to four decimal places.
o.4711
expected value
of a discrete random variable X is the mean of the random variable X. It is denoted E(X) and is given by computing the expression
the compliment
of an event A is the set of all outcomes in the sample space that are not in A
You guess that there is a 20% chance that you will be assigned homework in your Spanish class on Thursday
subjective probability
randomness
suggests a certain haphazardness of unpredictability. Randomness and uncertainty are both vague concepts that deal with variation. And even though these words are not synonyms, their discussion leads to the concept of probability.
Describe the complaint of the given event 66% of 19yr old males are at least 153 pounds
100%-66%=34
In how many ways can the letters in the word 'Alaska' be arranged
120
event
A subset of a sample space.
You have a stack of 5 textbooks: English, History, Statistics, Geology, and Psychology. How many ways can you arrange these textbooks on a shelf?
Because you can put each book on the bookshelf only once, you have five possible choices for the first book. Similarly, there are four possible choices for the second book, three possible choices for the third book, two possible choices for the fourth book, and only one book left for the fifth book. Using the Fundamental Counting Principle, we have 5x4x3x2x1=120 possibilities
5 girls and 5 boys. The rules state each child must kick the same number of times and alternate girl-boy or boy-girl. How many ways can a line-up be made for one round of kicking?
There are 10 possibilities for the first child, since it could be a boy first or a girl first. After this, there are 5 possibilities for the second child (the other gender). There are 4 choices for the 3rd child, 4 for the 4th child, 3 for the 5th child, 3 for the 6th child, 2 for the 7th child, 2 for the 8th child, 1 for the 9th child, and 1 for the 10th child:10(5)(4)(4)(3)(3)(2)(2)(1)(1) = 28,800
Most nonpersonalized license plates in the state of Utah consist of three numbers followed by three letters (excluding I, O, and Q). How many license plates are possible?
There are ten digits (0-9)possible for each of the first three characters. Likewise, there are 23 letters possible for the last three characters. Therefore, we have the following. 10x10x10x23x23x23=12167000 possibilities
toss a coin three times
There will be only one outcome since, for example, it is not possible to have exactly one and exactly two heads on the same trial. The outcome will be unknown before tossing the coin three times. The sample space can be specified and is composed of eight outcomes: S=S={TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}. This experiment meets the conditions of a random experiment. An event could be obtaining more than one head, which involves the following set of outcomes: {THH, HTH, HHT, HHH}.
To complete your holiday shopping, you need to go to the bakery, department store, grocery store, and toy store. If you are going to visit the stores in sequence, how many different sequences exist?
This is a permutation problem because the order in which you visit the stores matters. Note that there are 4 stores to visit. By the permutation definition there are 4!=24 sequences.
probability distribution
a description of how the probabilities are distributed over the values of the random variable
A phone usage study records the time spent on the phone each day for 189 randomly selected individuals.
continuous
The amount of time six randomly selected volleyball players play during a game.
continuous
At a local university, you poll a group of 95 students and find that 63 of them are from out-of-state.
relative frequency probability
The auto parts department of an automotive dealership sends out a mean of 7.3 special orders daily. What is the probability that, for any day, the number of special orders sent out will be exactly 3? Round your answer to four decimal places.
0.0438
What is the probability of rolling a sum of 10 on a standard pair of six-sided dice? Express your answer as a fraction or a decimal number rounded to three decimal places, if necessary.
0.0833
The auto parts department of an automotive dealership sends out a mean of 7.4 special orders daily. What is the probability that, for any day, the number of special orders sent out will be exactly 5? Round your answer to four decimal places
0.1130
Cars enter a car wash at a mean rate of 3 cars per half an hour. What is the probability that, in any hour, exactly 5 cars will enter the car wash? Round your answer to four decimal places.
0.1606
11 people in an office with 6 different phone lines. If all the lines begin to ring at once, how many groups of 6 people can answer these lines
462
probability law 2
A probability of one means the event must happen.
Choosing a three of spade out of a standard deck
Not mutually exclusive
An English teacher needs to pick 9 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 4 novels, 6 plays, 5 poetry books, and 5nonfiction books.Step 2 of 2 :If he wants to include all 4novels, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.
Total books =4+6+5+5=20 If he wants to include all 4 novels, then the number of books left to select = 9-4=5 Remaining choices for books = 20-4=16 4c4 x 16c5
combination
a grouping of items in which order does not matter
You believe you have a 12 chance of tossing a coin and getting a head
classical probability
Suppose that on the average, 6 students enrolled in a small liberal arts college have their automobiles stolen during the semester. What is the probability that less than 2 students will have their automobiles stolen during the current semester? Round your answer to four decimal places.
0.0174
What is the probability of rolling a sum of 1111 on a standard pair of six-sided dice? Express your answer as a fraction or a decimal number rounded to three decimal places, if necessary.
0.05556
The auto parts department of an automotive dealership sends out a mean of 6.8 special orders daily. What is the probability that, for any day, the number of special orders sent out will be exactly 3? Round your answer to four decimal places.
0.0584
A company produces optical-fiber cable with a mean of 0.7 flaws per 100 feet. What is the probability that there will be exactly 4 flaws in 1100 feet of cable? Round your answer to four decimal places.
0.0663
An experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 1 of 2 : What is the probability of getting a head on the coin and the number 1 on the die? Round your answer to four decimal places, if necessary
0.0833
What is the probability of getting a head on the coin and the number 1 on the die? Round your answer to four decimal places, if necessary.
0.0833
A company produces optical-fiber cable with a mean of 0.3 flaws per 100 feet. What is the probability that there will be exactly 4 flaws in 800 feet of cable? Round your answer to four decimal places.
0.1253
What is the probability that a customer is male and lives with his parents? Express your answer as a fraction or a decimal number rounded to four decimal places. A pizza delivery company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 1399 customers. Residence Males Females Apartment 81 228 Dorm 116 79 With Parent(s) 215 252 Sorority/Fraternity House 130 97 Other 129 72
0.1537
An experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 2 of 2 : What is the probability of getting a tail on the coin and at least a 5 on the die? Round your answer to four decimal places, if necessary.
0.1667
The number of calls received by an office on Monday morning between 8:00 AM and 9:00 AM has a mean of 5. Calculate the probability of getting no more than 4 calls between eight and nine in the morning. Round your answer to four decimal places.
0.4405
A well-mixed cookie dough will produce cookies with a mean of 5 chocolate chips apiece. What is the probability of getting a cookie with at least 4 chips? Round your answer to four decimal places
0.7350
Lost-time accidents occur in a company at a mean rate of 0.5 per day. What is the probability that the number of lost-time accidents occurring over a period of 8 days will be no more than 5? Round your answer to four decimal places.
0.7851 0.5*8 i.e. 4 per 8 days
Lost-time accidents occur in a company at a mean rate of 0.5 per day. What is the probability that the number of lost-time accidents occurring over a period of 10 days will be at least 3? Round your answer to four decimal places
0.8753
Suppose a basketball player has made 239 out of 358 free throws. If the player makes the next 22 free throws, I will pay you $12. Otherwise you pay me $8 Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
0.91
The number of calls received by an office on Monday morning between 8:00 AM and 9:00 AM has a mean of 2. Calculate the probability of getting no more than 4 calls between eight and nine in the morning. Round your answer to four decimal places.
0.9474
A well-mixed cookie dough will produce cookies with a mean of 7 chocolate chips apiece. What is the probability of getting a cookie with at least 3 chips? Round your answer to four decimal places.
0.9704
The number of weaving errors in a twenty-foot by ten-foot roll of carpet has a mean of 0.8. What is the probability of observing less than 4 errors in the carpet? Round your answer to four decimal places.
0.9909
The number of weaving errors in a twenty-foot by ten-foot roll of carpet has a mean of 0.8. What is the probability of observing less than 5 errors in the carpet? Round your answer to four decimal places
0.9986
An experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 2 of 2 : What is the probability of getting a tail on the coin and at least a 2 on the die? Round your answer to four decimal places, if necessary.
1. P(tail) = 0.5 P(at least 2) = 5/6 P(tail on the coin and at ;least 2 on the die) = 0.5 x 5/6 = 0.4167
A classmate walks into class and states that he has an extra ticket to a symphony orchestra concert on Friday night. He asks everyone in the class to put their name on a piece of paper and put it in a basket. He plans to draw from the basket to choose the person who will attend the concert with him. If there are 15 people in class that night, what is your chance of being chosen to attend the concert? Round your answer to four decimal places, if necessary.
1/15
A classmate walks into class and states that he has an extra ticket to a festival on Friday night. He asks everyone in the class to put their name on a piece of paper and put it in a basket. He plans to draw from the basket to choose the person who will attend the festival with him. If there are 15 people in class that night, what is your chance of being chosen to attend the festival? Round your answer to four decimal places, if necessary.
1/15 because 1 divided by total of ppl in the class
What is the probability of rolling a sum of 11 on a standard pair of six-sided dice? Express your answer as a fraction or a decimal number rounded to three decimal places, if necessary
1/18
in experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 2 of 2 : What is the probability of getting a tail on the coin and at least a 3 on the die? Round your answer to four decimal places, if necessary.
1/3
A classmate walks into class and states that he has an extra ticket to a classical music concert on Friday night. He asks everyone in the class to put their name on a piece of paper and put it in a basket. He plans to draw from the basket to choose the person who will attend the concert with him. If there are 30 people in class that night, what is your chance of being chosen to attend the concert? Round your answer to four decimal places, if necessary
1/30
A classmate walks into class and states that he has an extra ticket to a musical show on Friday night. He asks everyone in the class to put their name on a piece of paper and put it in a basket. He plans to draw from the basket to choose the person who will attend the show with him. If there are 30 people in class that night, what is your chance of being chosen to attend the show? Round your answer to four decimal places, if necessary
1/30
in how many ways can the letters in the word 'Coefficient' be arranged?
11! 2!2!2!2!1!1!1! 2494800
A coordinator will select 7 songs from a list of 13 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
13p7 = 8648640
DJ Joyce is making a playlist for an internet radio show; she is trying to decide what 12 songs to play and in what order they should be played. If she has her choices narrowed down to 19 hip-hop, 18 jazz, and 10 blues songs, and she wants to play an equal number of hip-hop, jazz, and blues songs, how many different playlists are possible? Express your answer in scientific notation rounding to the hundredths place
19c4=3876 18c4=3060 10c4=210 12p12=479001600 multiply all of these 1.193x10^18
You order some phone covers online and get an estimated delivery date of June 1-June 14. You know you will be out of town June 4-7 and are a little concerned about the package arriving when you are away. Assuming the delivery date follows a discrete uniform distribution, what is the likelihood your package will be delivered while you are out of town? Round your answer to four decimal places, if necessary.
2/7
You order some phone covers online and get an estimated delivery date of June 1-June 9. You know you will be out of town June 4th and 5th and are a little concerned about the package arriving when you are away. Assuming the delivery date follows a discrete uniform distribution, what is the likelihood your package will be delivered while you are out of town? Round your answer to four decimal places, if necessary.
2/9
If you throw exactly two heads in two tosses of a coin you win $28 If not, you pay me $13. Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
28*0.25=7 -13*0.25=-3.25 (this step is three times because there are three other options where you would lose money) =-2.75 if we played this game 994 times we would do -2.75*994 for a total lose of -2733.5
If a coin is tossed 6 times, and then a standard six-sided die is rolled 3 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?
2^6 6^3 52c3 305510400
You order some phone covers online and get an estimated delivery date of June 1-June 7. You know you will be out of town June 4th, 5th, and 6th and are a little concerned about the package arriving when you are away. Assuming the delivery date follows a discrete uniform distribution, what is the likelihood your package will be delivered while you are out of town? Round your answer to four decimal places, if necessary
3/7
find the number of outcomes when 158 books are nonfictional out of 310
310-158= 152
A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 3 types of drinks to choose from, 4 types of sandwiches, and 4 types of chips. How many different value meal packages are possible?
3x4x4=48
You are ordering a new home theater system that consists of a TV, surround sound system, and DVD player. You can choose from 8 different TVs, 16 types of surround sound systems, and 20 types of DVD players. How many different home theater systems can you build?
8x16x20=2560
Dave wants to buy a new collar for each of his 5 cats. The collars come in a choice of 9 different colors. How many selections of collars are possible if repetitions of colors are not allowed?
9x8x7x6x5=15120
Dave wants to buy a new collar for each of his 5 cats. The collars come in a choice of 9 different colors. Step 1 of 2 : How many selections of collars for the 5 cats are possible if repetitions of colors are allowed?
9x9x9x9x9=59049 there are 9 different options for 5 dogs
probability law 1
A probability of zero means the event cannot happen.
probability law 3
All probabilities must be between zero and one inclusively. That is, 0≤P(A)≤1
A well-mixed cookie dough will produce cookies with a mean of 7 chocolate chips apiece. What is the probability of getting a cookie with no more than 4 chips? Round your answer to four decimal places.
Let X denotes the number of chips in a randomly selected cookie. X ~ Poisson(7) The probability mass function of X is 0.173
When deciding what you want to put into a salad for dinner at a restaurant, you will choose one of the following extra toppings: mushrooms, asparagus. Also, you will add one of following meats: pepperoni, eggs. Lastly, you will decide on one of the following dressings: Italian, French.
MPI,MPF,MEI,MEF,API,APF,AEI,AEF
Determine whether or not the given procedure results in a binomial distribution. If not, identify which condition is not met. Surveying 83 people to determine which brand of apparel is their favorite.
No (since there can be more than 2 outcomes, therefore the condition that only 2 outcome for each trail is not followed)
An experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 1 of 2 : What is the probability of getting a head on the coin and the number 4 on the die? Round your answer to four decimal places, if necessary.
P(head on coin) = 1 / 2 P(Number 4 on die) = 1 / 6 P(head on coin AND Number 4 on die) = 1 / 2 * 1 / 6= 0.0833
Suppose a basketball player has made 231 out of 361 free throws. If the player makes the next 2 free throws, I will pay you $31. Otherwise you pay me $21. Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
P(winning) = (231/361)*(231/361) = 0.40946 P(loss) = 1-0.40946 = 0.59054 expected value of the proposition = (31*0.40946)+(-21*0.59054) = 0.29 (ans)
A classmate walks into class and states that he has an extra ticket to a musical show on Friday night. He asks everyone in the class to put their name on a piece of paper and put it in a basket. He plans to draw from the basket to choose the person who will attend the show with him. If there are 29 people in class that night, what is your chance of being chosen to attend the show? Round your answer to four decimal places, if necessary.
Probability of being chosen to attend the festival is computed as:= 1 / Total number of people in the class= 1/29
Write out the sample space for the given experiment. Use the letter R to indicate red, G to indicate green, and B to indicate blue. A die shows 3 different colors on it. Give the sample space for the next 2 rolls.
RR,RG,RB,GR,GB,GG,BB,BR,BG
At the Fidelity Credit Union, a mean of 5.5 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 5 customers will arrive? Round your answer to four decimal places.
Solution : Given that , mean = = 5.5 Using Poisson probability formula, P(X = x) = (e- * x ) / x! P(X > 5) = 1 - P(X 5) = 1 - 0.5289 = 0.4711 Probability = 0.4711
At the Fidelity Credit Union, a mean of 7.5 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 5 customers will arrive? Round your answer to four decimal place
Solution : Given that , mean = = 7.5 Using poisson probability formula, P(X = x) = (e- * x ) / x! P(X > ) = 1 - P(X 5 ) = 1 - ( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)) = 1 - ( (e-7.5 * 7.50) / 0! + (e-7.5 * 7.51) / 1! + (e-7.5 * 7.52) / 2! + (e-7.5 * 7.53) /3! + (e-7.5 * 7.54) / 4! + (e-7.5 * 7.55) / 5! ) = 1 - ( 0.0006 + 0.0041 + 0.0156 + 0.0389 + 0.0729 + 0.1094) = 1 - 0.2414 = 0.7586
probability law 4
The sum of the probabilities of all outcomes in a sample space must equal one. That is, if P(Ai) is the probability of outcome AiAi, and there are n such outcomes, then P(A1)+P(A2)+...+P(An)=1
Assume we have a deck of playing cards consisting of 13 hearts, 13 clubs, 13 spades, and 13 diamonds. Draw a card from a well-shuffled deck and observe the suit of the card. Have we met the three conditions of a random experiment?
There will be only one outcome. The suit will be unknown since the card will be drawn at random. The sample space consists of the following set of outcomes: S=S={heart, club, spade, diamond}. This experiment meets the conditions of a random experiment. If the random experiment involves drawing a card and observing a spade or a club, then the event would be given by the set of outcomes {spade, club}.
Toss a coin
There will only be one outcome for each trial of the experiment since it is not possible to observe both a head and a tail on the same toss. The outcome is unknown before the toss. The sample space can be specified and contains two outcomes, S={Head,Tail}S=HeadTail. Because each of the three conditions is satisfied, for all practical purposes, this experiment meets the conditions of a random experiment. It is possible for a coin to land on its edge and thus neither be heads nor tails. Any theory, however, involves a certain degree of idealization, which means, in this case, that landing on an edge will not be considered a possible outcome.
Inspect a transistor to determine if it meets quality control standards. Have we met the three conditions of a random experiment?
There will only be one outcome. The outcome of the experiment will be unknown if the transistor is selected from a manufacturing process that occasionally produces defective parts. The sample space consists of the set of outcomes, S=S={meets standards, does not meet standards}. This experiment meets the conditions of a random experiment.
Roll a die and observe the number of dots on the upper-most surface. Have we met the three conditions of a random experiment?
There will only be one outcome. The value of the outcome is not known. The sample space can be specified and is composed of outcomes, S={1,2,3,4,5,6}S=123456. This experiment meets the conditions of a random experiment. An example of an event could be rolling an even number, which would be given by the set of outcomes, {2,4,6}246
The number of weaving errors in a twenty-foot by ten-foot roll of carpet has a mean of 0.7. What is the probability of observing less than 5 errors in the carpet? Round your answer to four decimal places.
This is Poisson distribution Mean = m = 0.7 X = 5 P ( x < 5 ) = ( e -m * mx ) / x ! P ( x < 5 ) = 0.9992
At a local fast food restaurant, the door to the kitchen is secured by a five-button lock, labeled 1, 2, 3, 4, 5. To open the door, the correct three-digit code must be pushed but each button can only be pushed once. How many different codes are possible?
This is a permutation problem of 5 objects, but we are selecting only 3 at a time. There are5 buttons available for the first character in the code, 4 for the second, and 3 for the third. Therefore, there are (5)(4)(3)=60 possible codes.
Suppose that on the average, 3 students enrolled in a small liberal arts college have their automobiles stolen during the semester. What is the probability that more than 3 students will have their automobiles stolen during the current semester? Round your answer to four decimal places.
Thus , X follows a Poisson distribution with parameter The probability mass function of X is : 0.3528
If a coin is tossed 2 times, and then a standard six-sided die is rolled 3 times, and finally a group of two cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?
To obtain the solution to the problem, the number of possible outcomes for each task is computed and then the Fundamental Principle of Counting is applied to the three tasks. 2^2 outcomes possible when tossing a coin 2 times, 6^3 outcomes possible when rolling a standard six-sided die 3 times, and 52c2 outcomes possible when drawing two cards from a deck of cards without replacement. Applying the Fundamental Principle of Counting to these three tasks, we see that the total number of different outcomes possible is 1145664
The computer that controls a bank's automatic teller machine crashes a mean of 0.6 times per day. What is the probability that, in any seven-day week, the computer will crash more than 2 times? Round your answer to four decimal places.
We need to calculate P(X > 2) = 1 - P(X <= 2).P(X > 2) = 1 - (4.2^0 * e^-4.2/0!) + (4.2^1 * e^-4.2/1!) + (4.2^2 * e^-4.2/2!)P(X > 2) = 1 - (0.015 + 0.063 + 0.1323)P(X > 2) = 1 - 0.2103 = 0.7897
The computer that controls a bank's automatic teller machine crashes a mean of 0.3 times per day. What is the probability that, in any seven-day week, the computer will crash less than 4 times? Round your answer to four decimal places.
X ~ Poisson () Where = 0.3 per day For a week , = 0.3 * 7 = 2.1 Poisson probability distribution is P(X) = e-X / X! So, P(X < 4) = P( X <= 3) = P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) = e-2.1 2.10 / 0! +e-2.1 2.11 / 1! +e-2.1 2.12 / 2! +e-2.1 2.13 / 3! = 0.8386
The computer that controls a bank's automatic teller machine crashes a mean of 0.5 times per day. What is the probability that, in any seven-day week, the computer will crash more than 5 times? Round your answer to four decimal places.
X ~ Poisson () Where = 0.5 per day So for 7 days, = 0.5 * 7 = 3.5 Poisson probability distribution is P(x) = e-* x / x! So, P( X > 5) = 1 -P(x <= 5) = 1 - [ p(x= 0)+ p(x = 1) + p( x = 2) + p( x =3) + p ( x = 4) + p( x = 5) ] = 1 - [ e-3.5 + e-3.5 * 3.5 + e-3.5 * 3.52 / 2! + e-3.5 * 3.53 / 3! + e-3.5 * 3.54 / 4! + e-3.5 * 3.55 / 5! ] = 1 - 0.8576 = 0.1424
poisson random variable
an experiment must meet two conditions. Successes occur one at a time. (That is, two or more successes cannot occur at exactly the same point in time or exactly at the same point in space.) The occurrence of a success in any interval is independent of the occurrence of a success in any other interval
Classical probability
can be measured as a simple proportion: the number of outcomes that compose the event divided by the number of outcomes in the sample space, when it can be assumed that all of the outcomes are equally likely
discrete probability distribution
consists of all possible values of the discrete random variable along with their associated probabilities.
The amount of sugar imported by the U.S. in a day.
continuous
The number of words in your textbook.
discrete
The salaries of 73 randomly selected employee
discrete
A company produces optical-fiber cable with a mean of 0.3 flaws per 100 feet. What is the probability that there will be exactly 5 flaws in 1000 feet of cable? Round your answer to four decimal places.
expected number of flaws in 1000 feet =1000*0.3/100=3 therefore from poisson distribution probability that there will be exactly 5 flaws in 1000 feet of cable =e-3*35/5! =0.1008
Statistical inference
involves the use of sample data to form generalizations or inferences about a population. Using sample data to estimate the values of population parameters is one form of statistical inference
compound event
is an event that is defined by combining two or more events. A={annual income is greater than $50,000} and B={subscribes to more than one other sports magazine}.
Probability
likelihood that a particular event will occur
Choosing a student who is a physics major or chemistry major from a nearby university to participate in a research study. assume eacb student has one major
mutually inclusive
intersection
of the events A and B is the set of all outcomes that are included in both A and B. Symbolically, the intersection of A and B is denoted A∩B and is read " A intersect B."
union
of the events A and B is the set of outcomes that are included in event A or B or both, A∪BA∪B. Symbolically, the union of A and B is denoted A∪B and is read "A union B."
An experiment consists of tossing a coin and rolling a six-sided die simultaneously. Step 2 of 2 : What is the probability of getting a tail on the coin and at least a 3 on the die? Round your answer to four decimal places, if necessary.
p(a) 1/2 P (b) 4/6 1/2*4/6=1/3.
higher pay out Plan A Payout P( Payout ) -25000 0.13 45000 0.58 65000 0.29 Plan B Payout P( Payout ) -10000 0.24 25000 0.49 55000 0.27
plan a
Decide if the following probability is classical, relative frequency, or subjective. You calculate that the probability of randomly choosing a student who is living in the dorms is about 22%
relative frequency probability
You calculate that the probability of randomly choosing a student who is left-handed is about 35%
relative frequency probability
Poisson distribution
s similar to the binomial in that the random variable represents a count of the total number of successes. The major difference between the two distributions is that the Poisson does not have a fixed number of trials. Instead, the Poisson uses a fixed interval of time or space in which the number of successes are recorded. Thus, there is no theoretical upper limit on the number of successes, although large numbers of successes are not very likely.
You order some accessories online and get an estimated delivery date of June 1-June 13. You know you will be out of town June 4-7 and are a little concerned about the package arriving when you are away. Assuming the delivery date follows a discrete uniform distribution, what is the likelihood your package will be delivered while you are out of town? Round your answer to four decimal places, if necessary.
since out of 13 days , you are out of town for 4 days likelihood your package will be delivered while you are out of town =4/13
determine if the following value could be a probability 15/49
yes
