Knewton Alta Chapter 3 Probability Topics Part 1

event

a subset of the set of all outcomes of an experiment

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.

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.

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.

independent

Recall that if two events are independent that one occurring does not have any effect on the other occurring

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)?

Correct answers:$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.

Event: some subset of the possible outcomes of an experiment, can be described by listing the outcomes or described in words Outcome: any of the possible results of an experiment in probabilityOutcome is also known as Outcome space Trial: one repetition or instance of a repeated experiment

Event: some subset of the possible outcomes of an experiment, can be described by listing the outcomes or described in words Outcome: any of the possible results of an experiment in probabilityOutcome is also known as Outcome space Trial: one repetition or instance of a repeated experiment

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.

A standard six-sided die shows a number, 1, 2, 3, 4, 5, or 6, on each of its sides. You roll the die once. Let E be the event of rolling the die and it showing an even number on top and L be the event of rolling a number less than 4. Rolling a 3 is an outcome of which of the following events? Select all correct answers.

E OR L E′ AND L E′ OR L′ To roll a 3: It is an outcome of E′, that is NOT even. It is an outcome of L, less than 4. So, correct the only correct "AND" answer is: E′ AND L. There are many more correct "OR" answers: E′E′E∪L∪L′∪L

You have a standard deck of 52 cards. There are four suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards: Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King. Given Events A and B, are the two events mutually exclusive? Explain your answer. Event A: Drawing a 10. Event B: Drawing a heart.

No, the events are not mutually exclusive because they share the common outcome of 10 of hearts. 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 an outcome in common, 10 of hearts, so they are not mutually exclusive.

Is the statement below true or false? Mutually exclusive is the property of two events in which the occurrence of one of the events affects the chance of the other occurring.

False Mutually exclusive is defined as the property of events in which none can occur at the same time.

Given the following information, determine whether events B and C are independent, mutually exclusive, both, or neither. P(B)=56 P(B AND C)=12 P(C)=23

Neither Since P(B AND C)≠0, and P(B AND C)≠P(B)⋅P(C), the two events are neither independent nor mutually exclusive.

Explanation for these: Outcome: Event: Trial: Independent events: Dependent events: Mutually exclusive:

Outcome: any of the possible results of an experiment in probability. The Outcome aka Outcome space Event: some subset of the possible outcomes of an experiment, can be described by listing the outcomes or described in words Trial: one repetition or instance of a repeated experiment Independent events: events that have no influence on each other; if the fact that one event has occurred does not affect the probability that the other event will occur Dependent events: events that influence the occurrence of the other; if or not one event occurs does affect the probability that the other event will occur Mutually exclusive: events which are impossible to both occur or that have no outcomes in common. aka Disjoint events

Let R be the event that a randomly chosen person lives in the city of Raleigh. Let O be the event that a randomly chosen person is over 50 years old. Place the correct event in each response box below to show: Given that the person lives in Raleigh, the probability that a randomly chosen person is over 50 years old.

1$O$O​ 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 is over 50 years old given that the person lives in Raleigh, so the correct answer is P(O|R).

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.

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.

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 B mutually exclusive and are A and C independent?

A and B are mutually exclusive and A and C are independent.

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.

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?

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

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?

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

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.

Correct answer: 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).

The event of the weather being above 90∘ is A and the event of 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)?

Correct answers:$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.

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.

Correct answers:$\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=47/50.

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.

Correct answers:$\frac{17}{20}$17/20​​ 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 6/40. Therefore, by the complement rule, the probability of not choosing a multiple of 6 is 1−6/40=34/40=17/20.

mutually exclusive

Events that are mutually exclusive share no outcomes.

If P(A)=0.6 and P(B)=0.15, what can we say about the relationship between A and B?

In this situation, because we are not given P(A|B), P(B|A), or P(A AND B), we do not have enough information to decide if A and B are independent or mutually exclusive.

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).

Paul will roll two standard dice simultaneously. If Event A = both dice are odd and Event B = at least one die is even, which of the following best describes events A and B? Select two answers.

Mutually Exclusive Dependent Mutually exclusive events are events that cannot occur at the same time. In this case, both dice cannot be odd (event A) if at least one of the dice is even (event B). Independent events are those for the occurrence of one event has no effect on the probability of the other, and dependent events are any that are not independent. Mutually exclusive events are almost always also dependent (with the exception of events that are already impossible) since the occurrence of one event means the probability of the other event changes 0.

Beth is performing an experiment to check if a die is fair. She rolls the die 5 times and records the sequence of numbers she gets. Which of these is an outcome of this experiment? Select all correct answers.

Rolling the sequence 1,1,2,1,6 Rolling five 4's An outcome is a specific result of an experiment. So the outcomes of this experiment are all the possible sequences of five die rolls. So in this case, a particular sequence such as 1,1,2,1,6 is an outcome. So is rolling five 4's, because this is a specific outcome (4,4,4,4,4).

Given the following information, can we determine which pairs of A, B, and C are independent or dependent? P(A)=0.2 P(B)=0.5 P(C)=0.3 P(A|B)=0.5 P(B|C)=0.5 P(A|C)=0.2

Solution First, consider A and B. Note that P(A|B)=0.5≠0.2=P(A) So becauseP(A|B)≠P(A), we see thatAandBare not independent. In other words,AandBare dependent. Next, consider B and C. Note that P(B|C)=0.5=P(B) Therefore, becauseP(B|C)=P(B),BandCare independent. Finally, note that P(A|C)=0.2=P(A) So we see thatAandCare independent events by the first or second condition of independence given above.

In probability, we often use "OR," "AND," and "NOT" to describe events:

We say an outcome is in A OR B if it is in A, it is in B, or it is in both A and B. You will also see this denoted A∪B. We say an outcome is in A AND B if it is found in both A and B. You will also see this denoted A∩B. We say an outcome is NOTin A, and write A′ (A prime), if the outcome is not found in the event A. You may also see this written as ∼A.

Mutually Exclusive Events

are two or more events that cannot occur at the same time. Events that are Mutually Exclusive can also be described as Disjoint events.

A campus contains buildings numbered 1 through 70. What is the probability that a student will enter a building that is not a multiple of 12? Give your answer as a fraction. Reduce the fraction if necessary.

$\frac{13}{14}$1314​​ Note that there are 5 multiples of 12 in the range 1 to 70, namely 12, 24, 36, 48, 60. So the probability of the student entering one of these multiples is 570. Therefore, by the complement rule, the probability of not entering one of those multiples of 12 is 1−5/70=65/70=13/14.

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%

Correct answer: 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).

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.

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?

Correct answer: 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.

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

Correct answer: 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.

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)?

Correct answers:$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.

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

Correct answers:$\frac{23}{34}$23/34​​ 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.

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.

Correct answers:$\frac{7}{9}$7/9​​ 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 7/9.

A student works at an on-campus job Monday through Friday. The student also participates in intramural volleyball on Tuesdays and Thursdays. Given Events A and B, are the two events mutually exclusive? Explain your answer. Event A: On a random day of the week, the student is working at their on-campus job. Event B: On a random day of the week, the student is playing intramural volleyball.

No, the events are not mutually exclusive because they share the common outcomes of the student working and playing volleyball on certain days. 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 outcomes in common since the student both works and plays volleyball on Tuesdays and Thursdays. Thus, the events are not mutually exclusive.

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.

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).

Martin will draw 3 cards from a standard 52-card deck without replacement 5 different times. For each 3-card draw, he will record the number of red cards and the number of black cards. What is a trial of this experiment?

drawing 3 cards A trial is one specific execution of an experiment. In this case, each trial is a 3-card draw.

Conditional probability

the likelihood that an event will occur given that another event has already occurred

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).

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.

Which of the following best describes the term mutually exclusive?

the property of events in which none can occur at the same time Mutually exclusive is defined as the property of events in which none can occur at the same time.

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

Consider the Venn diagram below. Each of the dots outside the circle represents a graduating college student surveyed about their post-college job search. Five of the graduates are business majors: represented by red dots and labeled: 1,2,3,4,5 (abbreviated BM); Four of the graduates are social work majors: and are represented by blue dots: 1,2,3,4 (abbreviated SW). The two events represented in the Venn diagram are: Event A: The student has applied to at least one post-college job. Event B: The student currently is working in an internship position. Three of the business majors BM1,BM2,BM4 and two of the social work majors SW2,SW3 say they have applied to at least one job. Business majors BM2,BM3 state that they are currently working in an internship position. Social work majors SW2,SW4 say they are also currently working in an internship position. Students BM5,SW1 say that neither of the options currently apply to them. 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 an outcome: BM2,SW2. Thus, P(A and B)≠0 and the events are not mutually exclusive.

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′.

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 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

Correct answer: 425 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 425.

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?

Correct answer: 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 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?

Correct answer: 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.

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.

Correct answer: 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.

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

Correct answer: 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.

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.

Correct answer: 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 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?

Correct answer: 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.

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%

Correct answer: 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 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.

Correct answer: 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.

Correct answer: 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).

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.

Correct answer: 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).

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.

Correct answer: 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.

Correct answer: 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).

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.

Correct answer: 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.

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.

Correct answer: 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.

Which of the following gives the definition of event?

Correct answer: 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.

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?

Correct answer: 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.

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?

Correct answer: 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?

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

What is a particular result of an experiment?

Correct answer: outcome An outcome is defined as a particular result 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?

Correct answer: 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.

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)?

Correct answers:$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.

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)?

Correct answers:$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.

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.

Correct answers:$\frac{21}{26}$21/26​​ 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 5/26. The probability of the complement is 1−5/26=21/26.

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.

Correct answers:$\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.

Correct answers:$\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 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.

Correct answers:$\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 1−440=3640=9/10.

A card is drawn from a standard deck of 52 cards. Remember that a deck of cards has four suits: clubs, diamonds, hearts, and spades. Each suit has 13 cards: Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King. Given the three events, which of the following statements is true? Select all that apply. Event A: Drawing a clubs. Event B: Drawing a spade. Event C: Drawing a Queen.

Event A and Event B are mutually exclusive. 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 A is drawing a club and B is drawing a spade, so they have no outcomes in common. Therefore, A and B are mutually exclusive. C is drawing a Queen, which shares an outcome with A (Queen of Clubs) and with B (Queen of Spades). Therefore, C and A are not mutually exclusive, and C and B are not mutually exclusive.

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.

Two fair dice are rolled, one blue, (abbreviated B) and one red, (abbreviated R). Each die has one of the numbers {1,2,3,4,5,6} on each of its faces. The dots in the Venn diagram below show the number and the color of the dice. Let A be the event of rolling an even number on either of the dice. Let B be the event of rolling a number greater than 4 on either of the dice. 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 rolling an even number on either of the dice, so A should contain elements {R2,B2,R4,B4,R6,B6}. Event B is the event of rolling a number greater than 4 on either of the dice, so B should contain outcomes {R5,B5,R6,B6}. Event A AND B should therefore contain all outcomes that are greater than 4 AND even. Therefore, A AND B should contain the outcomes {R6,B6}. Notice that the outcomes R1, B1, R3, and B3 do not fall into either of these events. They should therefore be outside of the Venn diagram.

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 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 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.

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.

Subway trains and transit buses operate on different days and cannot occur together at the same time, even with delays. Given the following events, which of the statements is true? Select all that apply. Event A: The subway train "X" is delayed. Event B: The subway train "X" is on time. Event C: The transit bus "M" is delayed.

Event B and Event C are mutually exclusive. Event A and Event B are mutually exclusive. Event A and Event C are 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, A is a specific subway train being delayed and B is that same subway train being on-time. Therefore, A and B are mutually exclusive. We are given that subway trains and transit buses cannot occur together at the same time, even with delays. We know that event C is a transit bus being delayed. Then A and C are mutually exclusive. Also, B and C 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 fair die has six sides, with a number 1,2,3,4,5 or 6 on each of its sides. In a game of dice, the following probabilities are given: The probability of rolling two dice and both showing a 1 is 136; The probability of rolling the first die and it showing a 1 is 16; If you roll one die after another, the probability of rolling a 1 on the second die given that you've already rolled a 1 on the first die is 16. Let event A be the rolling a 1 on the first die and B be rolling a 1 on the second die. Are events A and B mutually exclusive, independent, neither, or both?

Events A and B are independent. Since P(B|A)=16, A and B are not mutually exclusive. However, A and B are independent since P(B|A)=P(B)=16.

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.

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.

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 deck of cards contains RED cards numbered 1,2,3,4,5 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 5 is one of the outcomes in which of the following events? Select all correct answers.

E′ R OR E R AND 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 E′, R AND O, and R OR E.

Independent events: events that have no influence on each other Dependent events: events that influence the occurrence of the other; Mutually Exclusive: events which are impossible to both occur or that have no outcomes in common; Mutually Exclusive events are also commonly referred to as Disjoint events. Conditional probability: the chance that an event will happen if another event has already happened.

Independent events: events that have no influence on each other Dependent events: events that influence the occurrence of the other; Mutually Exclusive: events which are impossible to both occur or that have no outcomes in common; Mutually Exclusive events are also commonly referred to as Disjoint events. Conditional probability: the chance that an event will happen if another event has already happened.

Mutually Exclusive Events

Mutually exclusive events are even easier to identify, because mutually exclusive events cannot happen simultaneously. Therefore, if A and B are mutually exclusive, P(A|B)=0 P(B|A)=0 P(A AND B)=0

Question Given the following information, what can we say about the relationship between events A and B? P(A)P(B)P(B|A)=0.21=0.53=0

Solution Because P(B|A)=0, we know that if A has occurred, then B cannot occur, which means that A and B are mutually exclusive. Note that A and B are dependent, because if they were independent then P(B|A) would equal P(B). Can two events A and B be mutually exclusive and independent? Let's think about what it would mean if they were. To be independent, it needs to be the case that P(A AND B)=P(A)⋅P(B) To be mutually exclusive, it needs to be the case that P(A AND B)=0 Putting these together, it must be that P(A)⋅P(B)=0, so either P(A)=0 or P(B)=0. This does not usually come up because we rarely deal with events that have probability zero, but it is still important to be aware of the definitions and where they lead.

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.

Say events A and B are independent. Then, P(A|B)=P(A), no matter what is going on with the P(B). This goes the other way as well. We can determine if two events are independent if we know information about P(A|B) and P(A)

Two or more events are said to be independent if the event of one occurring has no effect on whether or not the other one will also occur. We look for one of the following equivalent equations to determine independence: P(A|B)=P(A) P(B|A)=P(B) P(A AND B)=P(A)⋅P(B)

Which of the following pairs of events are independent? Select all correct answers.

You roll a die twice. Event A is getting an even number on the first roll. Event B is getting a 4 on the second roll. You flip a coin and roll a die. Event A is getting heads on the coin. Event B is getting a 3 or more on the die. The first correct answer is You roll a die twice. Event A is getting an even number on the first roll. Event B is getting a 4 on the second roll. What happens on the first roll does not affect what happens on the second roll. Therefore, knowing whether event A happened does not affect the probability of event B. So A and B are independent. The second correct answer is You flip a coin and roll a die. Event A is getting heads on the coin. Event B is getting a 3 or more on the die. What happens on the coin flip does not affect what happens on the roll of the die. Since event A only involves the coin and B only involves the die, the outcome of A does not affect the outcome of B. So A and B are independent.

Which of the following gives the definition of dependent?

the property of two events in which the occurrence of one of the events affects the chance of the other occurring Dependent is defined as the property of two events in which the occurrence of one of the events affects the chance of the other occurring.