alta- ch. 3

The event of the weather being above 90∘ is A and the eventof someone winning the lottery is B. If these events are independent events, using P(A)=0.26, and P(B)=0.84, what is P(A|B)?

$0.26$0.26​ Remember that because A and B are independent, knowing someone wins the lottery does not change the probability of the weather being above 90∘ . So P(A|B)=P(A)=0.26.

Storage units are numbered 1 through 50. What is the probability that the operator will choose a unit that is not a multiple of 8? Give your answer as a fraction. Reduce the fraction if necessary.

$\frac{22}{25}$2225​​ Note that there are 6 multiples of 8 in the range 1 to 50, namely 8, 16, 24, 32, 40, 48. So the probability that the operator will choose one of these multiples is 650. Therefore, by the complement rule, the probability of not choosing a multiple of 8 is 1−6/50=44/50=22/25.

A bag contains 11 RED beads, 10 BLUE beads, and 4 GREEN beads. If a single bead is picked at random, what is the probability that the bead is GREEN?

4/25 There are 4 green beads, and the total number of beads is 11+10+4=25. So the probability of getting a green bead is 4/25.

A deck of cards contains RED cards numbered 1,2,3,4,5, BLUE cards numbered 1,2, and GREEN cards numbered 1,2,3,4. If a single card is picked at random, what is the probability that the card is RED?

5/11 Because there are 5 red cards, and 11 cards total in the deck, the probability is 5/11.

A bag contains 10 RED beads, 3 BLUE beads, and 9 GREEN beads. If a single bead is picked at random, what is the probability that the bead is GREEN?

9/22 There are 9 green beads, and the total number of beads is 10+3+9=22. So the probability of getting a green bead is 9/22.

Consider the Venn diagram below. Each of the dots outside the circle represents a millionaire that has been questioned about their marital status. 8 millionaires are questioned. Six millionaires are men, represented by red dots and labeled M1,M2,M3,M4,M5,M6. Two millionaires are women, represented by blue dots and labeled W1,W2. The two events represented in the Venn diagram are: Event A: The millionaire is married. Event B: The millionaire is a man. The millionaires who said they were married are: M1,M2,M4,M5 and W1. The rest say they are not married. Place the dots in the appropriate event given the information above (you may not use all of the dots), then determine if the events are mutually exclusive.

A and B are not mutually exclusive. We say that two events are mutually exclusive if they share no outcomes, that is, P(A and B)=0. Here, A and B share outcomes: M1,M2,M4,M5. Thus, P(A and B)≠0 and the events are not mutually exclusive.

A deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.Drawing the Blue 1 is one of the outcomes in which of the following events? Select all correct answers.

B AND O R′ Because the card is blue and the number is odd, the card is an outcome of B and O. Therefore, it is also an outcome of B AND O and R′.

A deck of cards contains RED cards numbered 1,2 and BLUE cards numbered 1,2,3. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.Which of the following events include the outcome of drawing a blue 1?

B OR E B AND O Because the card is blue and the number is odd, the card is an outcome of the event B and of the event O. Therefore, it is also an outcome of the events B AND O and of the event B OR E.

Given the following information about events B and C P(C|B)=38 P(B)=12 P(C)=38 Are B and C mutually exclusive, independent, both, or neither?

B and C are independent because P(C|B)=P(C). Since P(C|B)=P(C)=38, we can conclude that B and C are independent. Because P(C|B)≠0, B and C are not mutually exclusive.

Given the following information about events B and C: P(B)=70% P(B AND C)=0 P(C)=45% Are B and C mutually exclusive, independent, both, or neither?

B and C are mutually exclusive because P(B AND C)=0. Since P(B AND C)=0, we can conclude that B and C are mutually exclusive. P(B AND C)≠P(B)⋅P(C), so we cannot conclude that the events are independent.

A deck of cards contains RED cards numbered 1,2, BLUE cards numbered 1,2,3, and GREEN cards numbered 1,2,3,4. If a single card is picked at random, what is the probability that the card is BLUE AND has an ODD number? Provide the final answer as a fraction.

Correct answers:$\frac{2}{9}$2/9​​ There are a total of 2+3+4=9 cards. Of these, there are 2 cards that are blue and have an odd number, so the answer is 2/9.

Pregnant women have the option of being scanned for Cystic Fibrosis risks in their unborn babies. If a mother or a father have a certain recessive gene, the baby is at risk for Cystic Fibrosis. Given the three events, which of the following statements is true? Select all that apply. Event A: The mother or father carries the recessive gene. Event B: The father carries the recessive gene. Event C: The baby is at risk for Cystic Fibrosis.

Event A and Event C are not mutually exclusive. Remember that in general, A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0. In this question, note that if the mother or father carries the recessive gene for cystic fibrosis, the baby is at risk. So here, A and C are not mutually exclusive, and B and C are not mutually exclusive. A and B are not mutually exclusive because they share an outcome: the father carrying the recessive gene.

A mathematics professor is organizing her classroom into groups for the final project. Each student will either be working on a graphing (G) project or writing a paper (P). Also, each student will be working on an economics (E), finance (F), sociology (S), or criminal justice (C) problem. The dots in the Venn diagram below show the different scenarios. Let A be the event of a student working on a graphing project. Let B be the event of a student writing a paper. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.

Event A is the event of a student working on a graphing project, so A should contain all outcomes with a G. Event B is the event of a student writing a paper, so B should contain all outcomes with a P. Event A AND B should therefore contain all outcomes with both a P and a G; however, none exist. Notice also that nothing should be outside of the Venn Diagram because every project is either a graphing project or a paper.

A CEO decides to award her employees that have met their objectives this year. Those employees that have met their objectives have the chance to win vacation days. They can win either Mondays (abbreviated M) or Tuesdays (abbreviated T). They can also win up to two days. The Venn Diagrams below show the different combinations that an employee can win. Let A be the event of winning two of the same day. Let B be the event of winning a Monday Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A\text{ AND } B. Note that you might not use all of the dots.

Event A is the event of winning two of the same day, so A should contain elements with repeated letters. Event B is the event of winning at least one Monday, so B should contain outcomes with the letter M . Event A AND B should therefore contain the outcome with two Mondays. Notice that the outcomes T does not fall into either of these events. It should therefore be outside of the Venn diagram.

Which of the following pairs of events are mutually exclusive? Select all correct answers. Recall in a 52 standard deck of cards, half the cards are red and half the cards are black. Also, face cards can be red or black.

Event A: rolling a 6-sided fair number cube and getting 1 Event B: rolling a 6-sided fair number cube and getting an even number Event A: drawing a card from a 52-card standard deck and getting a red face card Event B: drawing a card from a 52-card standard deck and getting a black card In the second answer choice, Event A is 1 and Event B is 2,4,6. Since the events have no outcomes in common, they are mutually exclusive. In the third answer choice, note that a card cannot be both red and black, so it cannot be a red face card and black, so Event A and Event B have no outcomes in common. Therefore, they are mutually exclusive.

Katy is deciding which charity to donate to. She is going to donate fifty dollars to each charity chosen, and she can donate to a women's shelter (abbreviated B), a charity for rescue animals (abbreviated R), and a children's foundation (C). The dots in the Venn diagram below show the combinations that she can donate to. Let X be the event of donating fifty dollars each to two charities for a total of one hundred dollars. Let Y be the event of donating to the charity for rescue animals. Move the dots on the Venn diagram to place the dots in the correct event, X, Y,or X AND Y. Note that you might not use all of the dots.

Event X is the event of donating one hundred dollars, which means 50 dollars to two charities, so X should contain outcomes BR, RC, and BC. Event Y is the event of donating to the charity for rescue animals, so Y should contain any outcome with an R. Event X AND Y should therefore contain all outcomes that have two charities listed and an R. Therefore, X AND Y should contain the outcome BR and RC. Notice that the outcomes B and C do not fall into either of these events. They should therefore be outside of the Venn diagram.

A game requires that players draw a blue card and red card to determine the number of spaces they can move on a turn. Let A represent drawing a red card, with four possibilities 1,2,3, and 4. Let B represent drawing a blue card, and notice that there are three possibilities 1,2, and 3. If the probability of a player drawing a red 2 on the second draw given that they drew a blue 2 on the first draw is P(R2|B2)=14, what can we conclude about events A & B?

Events A and B are independent since P(R2|B2)=P(R2). We are given that P(R2|B2)=14 and we can find that P(R2)=14 by finding the probability of drawing a red 2. Then, P(R2|B2)=P(R2)=14 and we can conclude that the events are independent.

Doctors conducting a pharmaceutical study release the following information: The probability that a patient will receive a trial medication in the first round is 1140; The probability that a patient will receive a placebo medication in the second round given that they received a trial medication in the first round is 13; The probability that a patient will receive a placebo medication in the second round is 710. Let event A be the event that a patient receives a trial medication in the first round and event B be the event that a patient receives a placebo medication in the second round. Are events A and B mutually exclusive, independent, both, or neither?

Events A and B are neither mutually exclusive nor independent. The correct conclusion is that the events are neither mutually exclusive nor independent. Since P(B|A)≠0, the events are not mutually exclusive. Because P(B|A)≠P(B), the events are not independent.

A deck of cards contains RED cards numbered 1,2,3,4,5,6 and BLUE cards numbered 1,2,3,4,5. Let: R be the event of drawing a red card, B be the event of drawing a blue card, E be the event of drawing an even numbered card, and O be the event of drawing an odd card. Drawing the Blue 3 is one of the outcomes in which of the following events?

E′ B AND O Because the card is blue and the number is odd, the card is an outcome of B and O. Since the card is not even, it is an element of E′ (remember that E′ is the complement of E).Therefore, it is also an outcome of B AND O and E′.

A restaurant is offering chicken specials numbered 1,2,3,4,5,6 and fish specials numbered 1,2,3,4,5. Let: C be the event of selecting a chicken special, F be the event of selecting a fish special, E be the event of selecting an even numbered special, and O be the event of selecting an odd special. Selecting the fish special number 3 is one of the outcomes in which of the following events? Select all that apply.

E′ F AND O Because the special is fish and the number is odd, the selection is an example of F and O. Since the selection is not odd, it is an element of E′ (remember that E′ is the complement of E).Therefore, it is also an example of C AND O and E′.

Let C be the event that a randomly chosen cancer patient has received chemotherapy. Let E be the event that a randomly chosen cancer patient has received elective surgery. Identify the answer which expresses the following with correct notation: Of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery.

P(E|C) Remember that in general, P(A|B) is read as "The probability of A given B," or equivalently, as "Of all the times B occurs, the probability that A occurs also." So in this case, the phrase "Of all the cancer patients who have received chemotherapy" can be rephrased to mean "Given that a cancer patient has received chemotherapy," so the correct answer is P(E|C).

Let S be the event that a randomly chosen voter supports the president. Let W be the event that a randomly chosen voter is a woman. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen voter is a woman, given that the voter supports the president.

P(W|S) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the voter supports the president, so the correct answer is P(W|S).

A deck of cards contains RED cards numbered 1,2,3,4 and BLUE cards numbered 1,2,3, as shown below. Let R be the event of drawing a red card, B be the event of drawing a blue card, E be the event of drawing an even numbered card, and O be the event of drawing an odd numbered card. Drawing the Red 3 is an outcome in which of the following events? Select all correct answers.

R AND O E′ E OR R The Red 3 is both red and odd, so it is an outcome in both R and O. Therefore, it is an outcome in R AND O. The Red 3 is not even, so it is an outcome in E′. The Red 3 is an outcome in R, so it is in E OR R (even though it is not in E).

A college offers its students multiple study-abroad opportunities for the semester. Given Events A and B, are the two events mutually exclusive? Explain your answer. Event A: The same student chooses to study abroad in Asia. Event B: The same student chooses to study abroad in Europe.

Yes, the events are mutually exclusive because they have no outcomes in common. A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0. In this case, A and B have do not have an outcome in common, so they are mutually exclusive.

A deck of cards contains RED cards numbered 1,2,3, BLUE cards numbered 1,2, and GREEN cards numbered 1,2,3,4,5,6. If a single card is picked at random, what is the probability that the card has an EVEN number?

5/11 By counting, we can see that there are 5 even cards, and a total of 11 cards in the deck. So the probability is 5/11.

A deck of cards contains RED cards numbered 1,2,3, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2. If a single card is picked at random, what is the probability that the card has an ODD number?

5/9 By counting, we can see that there are 5 odd cards, and a total of 9 cards in the deck. So the probability is 5/9.

A deck of cards contains RED cards numbered 1,2,3,4,5,6, BLUE cards numbered 1,2,3,4,5, and GREEN cards numbered 1,2,3. If a single card is picked at random, what is the probability that the card is RED?

6/14 Because there are 6 red cards, and 14 cards total in the deck, the probability is 6/14.

A game show releases its secrets about how it chooses contestants from its audience. They say The probability of being chosen for the first round on the show is 130; The probability of being chosen for the second round on the show is 215; The probability of being chosen for both rounds on the show is 0. Let event A be being chosen for the first round, and event B being chosen for this second round. Are events A and B mutually exclusive, independent, both, or neither?

Events A and B are mutually exclusive. Since P(A AND B)=0, A and B are mutually exclusive. Because P(A AND B)≠P(A)⋅P(B), A and B are not independent.

Which of the following gives the definition of event?

a subset of the set of all outcomes of an experiment An event is defined as a subset of the set of all outcomes of an experiment.

Patricia will draw 8 cards from a standard 52-card deck with replacement. Which of the following are not events in this experiment?

drawing 1 card Events are any combinations of outcomes or particular results in an experiment. Drawing 8 cards is the experiment and drawing 1 card is a trial of the experiment, neither of which specify a result or outcome.

Patricia will draw 8 cards from a standard 52-card deck with replacement. Which of the following are not events in this experiment?

drawing 1 card Events are any combinations of outcomes or particular results in an experiment. Drawing 8 cards is the experiment and drawing 1 card is a trial of the experiment, neither of which specify a result or outcome.

Using a standard 52-card deck, Michelle will draw 6 cards with replacement. If Event A = drawing all hearts and Event B = drawing no face cards, which of the following best describes events A and B?

independent Events are independent if the knowledge of one event occurring does not affect the chance of the other event occurring. In this case, the chance of either event occurring is the same whether or not the other event occurs.

Trial best fits which of the following descriptions?

one repetition or instance of an experiment A trial is defined as one repetition or instance of an experiment.

Arianna will roll a standard die 10 times in which she will record the value of each roll. What is a trial of this experiment?

one roll of the die A trial is one specific execution of an experiment. In this case, each trial is one roll of the die.

Which of the following gives the definition of trial?

one specific execution of an experiment A trial is defined as one specific execution of an experiment.

A fair spinner contains the numbers 1, 2, 3, 4, and 5. For an experiment, the spinner will be spun 5 times. If Event A = the spinner lands on numbers all less than 3, what is an outcome of Event A?

spinner lands on 1, 2, 1, 2, 2 An outcome of an event is any way in which the event can occur. In this case, the only option that results in Event A is the spinner landing on 1, 2, 1, 2, 2.

Charities at an auction are numbered 1 through 80. What is the probability that the charity chosen is not a multiple of 12? Give your answer in fraction form. Reduce the fraction if necessary.

$\frac{37}{40}$3740​​ Note that there are 6 multiples of 12 in the range 1 to 80, namely 12, 24, 36, 48, 60, 72. So the probability of the charity choosing one of these multiples is 680. Therefore, by the complement rule, the probability of not choosing a multiple of 12 is 1−6/80=74/80=37/40

A deck of cards contains RED cards numbered 1,2,3,4,5, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2,3,4,5,6. If a single card is picked at random, what is the probability that the card is GREEN?

6/15 Because there are 6 green cards, and 15 cards total in the deck, the probability is 6/15.

Given the following information, determine whether events B and C are independent, mutually exclusive, both, or neither. P(B)=0.6 P(B AND C)=0 P(C)=0.4 P(B|C)=0

Mutually Exclusive B and C are mutually exclusive because P(B AND C)=0 and P(B|C)=0. Events are mutually exclusive if they share no common outcomes.

The event of you going to work is A and the event of you taking leave is B. If these events are mutually exclusive events, using P(A)=0.55, and P(B)=0.10, what is P(A|B)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible for you to go to work if you take leave. Therefore, P(A|B)=0.

The event you walk to work on Monday is A and the event you drive to work on Monday is B. If these events are mutually exclusive events, using P(A)=0.22, and P(B)=0.62, what is P(B|A)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible for you to drive to work on Monday if you walk to work on Monday. Therefore, P(B|A)=0.

An HR director numbered employees 1 through 50 for an extra vacation day contest. What is the probability that the HR director will select an employee who is not a multiple of 13? Give your answer as a fraction. Reduce the fraction if necessary.

$\frac{47}{50}$4750​​ Note that there are 3 multiples of 13 in the range 1 to 50, namely 13, 26, 39. So the probability of the HR director selecting one of these multiples is 350. Therefore, by the complement rule, the probability of not selecting a multiple of 13 is 1−350=4750.

Given the following information about events A and B P(A)=0 P(A AND B)=0 P(B)=0.25 Are A and B mutually exclusive, independent, both, or neither?

A and B are both independent and mutually exclusive. A and B are independent because P(A AND B)=P(A)⋅P(B). A and B are also mutually exclusive because P(A AND B)=0. Thus, we conclude that A and B are both mutually exclusive and independent.

A motor company is manufacturing pickup trucks, sedans, minivans, SUVs, ATV, and motorcycles. The dots in the Venn diagram below show the type of each vehicle. A vehicle is selected at random. Let A be the event of selecting a four-wheeled vehicle. Let B be the event of selecting a pickup truck or a motorcycle. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots. Note: ATVs are four-wheeled vehicles.

Event A is the event of selecting a vehicle with four wheels, so A should contain the pickup truck, sedan, ATV, minivan, and the SUV. Event B is the event of selecting a pickup truck or a motorcycle. Event A AND B should therefore contain a pickup truck because it has four wheels.

There are 26 cards in a hat, each of them containing a different letter of the alphabet. If one card is chosen at random, what is the probability that it is not between the letters L and P, inclusive? Write your answer in fraction form. Reduce the fraction if necessary.

$\frac{21}{26}$2126​​ Use the complement rule to find this probability. The complement of this event is the event that the card chosen has a letter from L to P. There are 5 of these letters so the probability of drawing a card with one of these letters is 526. The probability of the complement is 1−5/26=21/26.

The event you wear flats to school today is A and the event you wear sneakers to school today is B. If these events are mutually exclusive events, using P(A)=0.34, and P(B)=0.26, what is P(B|A)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible for you to wear sneakers to school today if you wear flats to school today. Therefore, P(B|A)=0.

Phones collected from a conferences are labeled 1 through 40. What is the probability that the conference speaker will choose a number that is not a multiple of 6? Give your answer as a fraction. Reduce the fraction if necessary.

$\frac{17}{20}$1720​​ Note that there are 6 multiples of 6 in the range 1 to 40, namely 6, 12, 18, 24, 30, 36. So the probability of the speaker choosing one of these multiples is 640. Therefore, by the complement rule, the probability of not choosing a multiple of 6 is 1−6/40=34/40=17/20.

Let M be the event that a randomly chosen student passes a math test. Let S be the event that a randomly chosen student studies every day. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen student studies every day, given that the student passes a math test.

P(S|M) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the student passes a math test, so the correct answer is P(S|M).

The event of using your free hour to nap is A and the event of using your free hour to study is B. If these events are mutually exclusive events, using P(A)=0.23, and P(B)=0.73, what is P(B|A)?

Remember that because A and B are mutually exclusive, it is impossible for you to use your free hour to study if you use your free hour to nap. Therefore, P(B|A)=0.

The event of eating breakfast at a diner is A and the event of watching cable is B. If these events are independent events, using P(A)=0.22, and P(B)=0.46, what is P(B|A)?

$0.46$0.46​ Remember that because A and B are independent, knowing that someone eats breakfast at a diner does not change the probability of someone watching cable. So P(B|A)=P(B)=0.46.

Given the following information about events A and B: P(A)=112 P(A AND B)=116 P(A|B)=112 Are events A and B mutually exclusive, independent, both, or neither?

A and B are independent since P(A|B)=P(A). Since P(A|B)=112=P(A), we can conclude that events A and B are independent. They are not mutually exclusive because P(A|B)≠0. Your answer: A and B are mutually exclusive since P(A|B)=P(A). Incorrect. P(A|B)=P(A) here, but this does not mean the events are mutually exclusive. It means they are independent.

Consider the Venn diagram below. Each of the dots outside the circle represents a university that was polled. In this experiment, 8 universities were asked about their student demographics. Five of the universities are public universities: represented by red dots and labeled: 1,2,3, 4, 5 (abbreviated U); Three of the universities are private universities: and are represented by blue dots: 1,2, 3 (abbreviated P). The two events represented in the Venn diagram are: Event A: The majority of the university's students are women. Event B: The majority of the university's students are non-traditional college students. Universities U1,U2,U3, and P1 say that the majority of their students are women. Universities P2 and P3 say that the majority of their students are non-traditional students. Place the dots in the appropriate event given the information above (you may not use all of the dots), then determine if the events are mutually exclusive.

A and B are mutually exclusive. We say that two events are mutually exclusive if they share no outcomes, that is, P(A and B)=0. Here, A and B do not share any outcomes. Thus, P(A and B)=0 and the events are mutually exclusive.

Given the following information about events A, B, and C.​ P(A)P(B)P(C)=0.62=0.34=0.07P(B|A)P(C|B)P(A|C)=0=0.34=0.62 Are A and C mutually exclusive, independent, both, or neither?

A and C are independent because P(A|C)=P(A). P(A|C)=P(A)=0.62, so we should conclude that A and C are independent.

A deck of cards contains RED cards numbered 1,2,3 and BLUE cards numbered 1,2,3,4. Let R be the event of drawing a red card, B the event of drawing a blue card, E the event of drawing an even numbered card, and O the event of drawing an odd card.Drawing the Red 1 is one of the outcomes in which of the following events? Select all correct answers.

B′ R OR E B OR O Because the card is red and the number is odd, the card is an outcome of R and O. Therefore, it is also an outcome of B′, R OR E, and B OR O.

A university offers finance courses numbered 1,2,3,4,5 and accounting courses numbered 1,2,3,4,5,6. Let F be the event of selecting a finance course, A the event of selecting an accounting course, E the event of selecting an even numbered course, and O the event of selecting an odd course.Selecting the accounting course number 3 is one of the outcomes in which of the following events? Select all correct answers.

Correct answer: A AND O F OR O E′ Because the course is accounting and the number is odd, the course is an outcome of A and O. Therefore, it is also an outcome of A AND O, E′, and F OR O.

Different types of advertising methods are being considered for a company's new product: a magazine ad (M), a television ad (T), a newspaper coupon (N), a radio ad (R), a coupon mailer (C), and a social media ad (S). The coupons are both good for ten dollars off the item. The dots in the Venn diagram below show the various methods. An advertising specialist considers a method at random to review. Let A be the event of selecting a method that gives a discount. Let B be the event of selecting a printed method. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.

Event A is the event of selecting a method that gives a discount, so this includes both types of coupons. Event B is the event of selecting a printed method, so this includes the newspaper coupon, magazine ad, and coupon mailer. Event A AND B should therefore contain all printed methods that give a discount, so this includes the newspaper coupon and coupon mailer, which is all of event A. Notice that the social media ad, radio ad, and television ad do not fall into either of these events. They should therefore be outside of the Venn diagram.

A biologist has a number of butterfly specimens. The butterflies are of various colors and various ages. The colors are green (abbreviated G), red (abbreviated R), or yellow (abbreviated Y). Each specimen is labeled with one the numbers {1,2,3,4,5,6} and the number represents how many months old it is. The dots in the Venn diagram below show the age and the color of the specimen. The biologist selects a specimen at random. Let A be the event of selecting a yellow specimen. Let B be the event of selecting a specimen that is older than 2 months old. Move the dots on the Venn diagram to place the dots in the correct event, A, B,or A AND B. Note that you might not use all of the dots.

Event AA is the event of selecting a yellow butterfly, so AA should contain all butterflies with a YY. Event B is the event of selecting a butterfly older than two months, so B contains all butterflies labeled with a 3 or higher for any color. Event A \text{ AND } B should therefore contain all outcomes that are greater than 2 AND yellow. Therefore, A \text{ AND } B does not contain any outcomes. Notice that the outcomes 1 and 2 on green and red butterflies do not fall into either of these events. They should therefore be outside of the Venn diagram.

Given the following information, determine whether events B and C are independent, mutually exclusive, both, or neither. P(B)=0.35 P(B AND C)=0 P(B|C)=0

Mutually Exclusive B and C are mutually exclusive because P(B AND C)=0 and P(B|C)=0. They are not independent since P(B|C)≠P(B). Events are mutually exclusive if they share no common outcomes.

Given the following information, determine whether events B and C are independent, mutually exclusive, both, or neither. P(B)=60% P(B AND C)=0 P(C)=85%

Mutually Exclusive B and C are mutually exclusive because P(B AND C)=0. B and C are not independent because P(B AND C)≠P(B)⋅P(C).

Let B be the event that a randomly chosen person has low blood pressure. Let E be the event that a randomly chosen person exercises regularly. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person exercises regularly, given that the person has low blood pressure.

P(E|B) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the person has low blood pressure, so the correct answer is P(E|B).

Let G be the event that a randomly chosen employee of a restaurant is a General Manager. Let S be the event that a randomly chosen employee of a restaurant works at a seafood restaurant. Identify the answer which expresses the following with correct notation: Given that the employee is a General Manager, the probability that a randomly chosen employee of a restaurant works at a seafood restaurant.

P(S|G) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the employee is a General Manager, so the correct answer is P(S|G).

Suppose that A is the event you purchase an item from an online clothing store, and B is the event you purchase the item from a nearby store. If A and B are mutually exclusive events, P(A)=0.57, and P(B)=0.17, what is P(A|B)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible for the item to be purchased both online and at the nearby store. Therefore, P(A|B)=0.

The event the number 9 car wins the race is A and the event the number 8 car wins the race is B. If these events are mutually exclusive events, using P(A)=0.39, and P(B)=0.31, what is P(A|B)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible for the number 9 car to win if the number 8 car wins. Therefore, P(A|B)=0.

In New York City, the event the weather will be 82∘ today is A and the event the weather is 45∘ today is B. If these events are mutually exclusive events, using P(A)=0.29, and P(B)=0.25, what is P(B|A)?

$0$0​ Remember that because A and B are mutually exclusive, it is impossible today for the weather in New York City to be 45∘ if the weather in New York City will be 82∘. Therefore, P(B|A)=0.

The event you play football is A and the event your dog visits the vet is B. If these events are independent events, using P(A)=0.63, and P(B)=0.06, what is P(A|B)?

$0.63$0.63​ Remember that because A and B are independent, knowing your dog visits the vet does not change the probability of you playing football. So P(A|B)=P(A)=0.63.

Getting selected as class secretary is A and having pizza for lunch is B. If these events are independent events, using P(A)=0.70, and P(B)=0.67, what is P(A|B)?

$0.70$0.70​ Remember that because A and B are independent, having pizza for lunch does not change the probability of getting selected as class secretary. So P(A|B)=P(A)=0.70.

The event that your uncle calls you on a Sunday is A and the event your sister calls you on a Tuesday is B. If these events are independent events, using P(A)=0.42, and P(B)=0.83, what is P(B|A)?

$0.83$0.83​ Remember that because A and B are independent, knowing that your uncle calls you on a Sunday does not change the probability of your sister calling you on a Tuesday. So P(B|A)=P(B)=0.83.

Pictures from an artist are numbered 1 through 60. What is the probability that the artist will chose a picture that is not a multiple of 13? Give your answer as a fraction. Reduce the fraction if necessary

$\frac{14}{15}$1415​​ Note that there are 4 multiples of 13 in the range 1 to 60, namely 13, 26, 39, 52. So the probability of the artist choosing one of these multiples is 460. Therefore, by the complement rule, the probability of not choosing a multiple of 13 is 1−4/60=56/60=14/15.

On a fair, 8-sided die, each face is painted as follows: the 1 and 2 are painted green, 3 through 6 are painted blue, 7 is painted yellow, and 8 is painted gray. If the die is rolled one time, what is the probability that the roll will show an even number AND a face that is painted blue? Provide the final answer as a fraction.

$\frac{1}{4}$1/4​​ There are a total of 8 faces on the die, and thus 8 possible outcomes. There are 4 even numbers, namely 2,4,6 and 8, and there are 4 sides that are blue, namely 3,4,5, and 6. The only faces that feature an even number AND are painted blue are the blue 4 and the blue 6, so the answer is 2/8, or 1/4.

A bag contains 35 marbles, 11 of which are red. A marble is randomly selected from the bag, and it is blue. This blue marble is NOT placed back in the bag. A second marble is randomly drawn from the bag. Find the probability that this second marble is NOT red. Provide the final answer as a fraction.

$\frac{23}{34}$2334​​ Initially, 11 of the 35 marbles are red. That means that 35−11=24 of the marbles are NOT red. Once the blue marble is removed, the bag contains only 34 total marbles, but all 11 red marbles are still in the bag. Thus there are now 34−11=23 marbles that are NOT red. Thus, the probability that the second marble is NOT red is 23/34.

Seventy cards are numbered 1 through 70, one number per card. One card is randomly selected from the deck. What is the probability that the number drawn is a multiple of 3 AND a multiple of 5? Enter your answer as a simplified fraction.

$\frac{2}{35}$235​​ From the first 70 natural numbers, there are 23 multiples are 3: 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66, and 69. There are 14 multiples of 5: 5,10,15,20,25,30,35,40,45,50,55,60,65, and 70. The intersection of these two sets contains the four numbers: 15,30,45, and 60. Thus, the probability that the number is a multiple of 3 AND a multiple of 5 is 470, which simplifies to 235. ALTERNATIVE SOLUTION: Any number that is a multiple of both 3 and 5 is a multiple of 3⋅5=15. Thus, one can simply count the multiples of 15 that are less than or equal to 70. Those numbers are 15,30,45, and 60, which brings us to the same conclusion of 470, or 235.

A single card is randomly drawn from a standard 52-card deck. Find the probability that the card is a face card AND is red.(Note: aces are not generally considered face cards, so there are 12 face cards. Also, a standard deck of cards is half red and half black.) Provide the final answer as a fraction.

$\frac{3}{26}$326​​ There are 4 suits in a standard deck of cards, each suit containing 3 face cards (the jack, queen, and king). Thus, there are a total of 3⋅4=12 face cards. Of these 12 cards, half are red, which means there are 12÷2=6 red face cards. Since the entire deck contains 52 cards, the answer is 652, or 326. Another approach to this problem is to list each of the 6 red face cards: the jack, queen, and king of hearts, and the jack, queen, and king of diamonds. Again, since the deck contains a total of 52 cards, probability of drawing a red face card is 652, or 326.

A deck of cards contains RED cards numbered 1,2,3, BLUE cards numbered 1,2,3,4, and GREEN cards numbered 1,2. If a single card is picked at random, what is the probability that the card is BLUE OR has an ODD number? Provide the final answer as a fraction.

$\frac{7}{9}$79​​ There are a total of 3+4+2=9 cards. Of these, there are 4 blue cards, 2 red odd cards, and 1 green odd card. Thus the total number of cards we are interested in is 7, so the answer is 79.

A moving company has boxes numbered 1 through 40. What is the probability that the first box chosen by a mover is not a multiple of 9? Give your answer as a fraction. Reduce the fraction if necessary.

$\frac{9}{10}$910​​ Note that there are 4 multiples of 9 in the range 1 to 40, namely 9, 18, 27, 36. So the probability of the mover picking a box of one of these multiples is 440. Therefore, by the complement rule, the probability of not picking a box numbered with a multiple of 9 is 1−4/40=36/40=9/10.

Let R be the event that a randomly chosen person has red hair. Let G be the event that a randomly chosen person has green eyes. Place the correct event in each response box below to show: Given that the person has red hair, the probability that a randomly chosen person has green eyes.

1$G$G​ 2$R$R​ Remember that in general, P(A|B) is read as "The probability of A given B". Here we want to know the probability that a person has green eyes given that the person has red hair, so the correct answer is P(G|R).

A bag contains 11 RED beads, 3 BLUE beads, and 5 GREEN beads. If a single bead is picked at random, what is the probability that the bead is BLUE?

3/19 There are 3 blue beads, and the total number of beads is 11+3+5=19. So the probability of getting a blue bead is 3/19.

The following information about undergraduates at a university is given: The probability that a randomly chosen undergraduate attends office hours is 24%; The probability that a randomly chosen undergraduate has a 3.30 GPA or higher is 15.5%; The probability that a random chosen undergraduate has a 3.30 GPA or higher given that they attend office hours is 12%. Let event A is an undergraduate attending office hours and event B is an undergraduate having a 3.30 or higher GPA. Are events A and B mutually exclusive, independent, both, or neither?

Events A and B are neither independent nor mutually exclusive. Since P(B|A)≠0, we cannot conclude that A and B are mutually exclusive. Because P(B|A)≠P(B), A and B are not independent.

Given the following information, determine whether events B and C are independent, mutually exclusive, both, or neither. P(B|C)=22% P(B)=22% P(C)=17%

Independent P(B|C)=P(B)=22%, so the two events are independent. However, since P(B|C)≠0, the two events are not mutually exclusive.

Let S be the event that a randomly chosen store is having a sale. Let M be the event that a randomly chosen store has marked up their prices in the last six months. Identify the answer which expresses the following with correct notation: Given that the store is having a sale, the probability that a randomly chosen store has marked up their prices in the last six months.

P(M|S) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the store is having a sale, so the correct answer is P(M|S).

On an Alaskan cruise, shore excursions are offered most days. One day's options were: Kayaking to a glacier; Hiking to a waterfall. Cruise travelers can choose to participate in one excursion,or no excursions. A family on the cruise is divided on which activity to choose. Jack and Shirley want to kayak. Emma and Chris want to hike to a waterfall. Kelly wants to stay on the ship and read her book. Arrange the family members in their activity choice for the day in the Venn diagram below. Then, use the Venn diagram to answer the question: Are kayaking and hiking mutually exclusive events?

The kayaking and hiking shore excursion options share no participants, so they are mutually exclusive. Without the Venn diagram, we knew that cruise travelers had to choose one, or none, of the shore excursions. So, when someone chooses to kayak, they can't also hike. Therefore, there will be no shared outcomes.

At a major international airport, passengers are questioned about their destination. Given Events A and B, are the two events mutually exclusive? Explain your answer. Event A: The passengers are traveling to Paris, France. Event B: The passengers are NOT traveling to Paris, France.

Yes, the events are mutually exclusive because they have no outcomes in common. A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B)=0. In this case, A and B do not have an outcome in common, so they are mutually exclusive. Your answer: Yes, the events are mutually exclusive because P(A) is not equal to P(B). This is not part of the definition of mutually exclusive events.

Jacqueline will spin a fair spinner with the numbers 0, 1, 2, 3, and 4 a total of 3 times. If Event A = spinner lands on numbers all greater than 2 and Event B = total sum of 9, which of the following best describes events A and B?

dependent Events are independent if the knowledge of one event occurring does not affect the chance of the other event occurring. In this case, the chance of Event B occurring is greater if Event A occurs, so these events are dependent.

What is a particular result of an experiment?

outcome An outcome is defined as a particular result of an experiment.