Question 1

A short quiz has two true-false questions and one multiple-choice question with four choices. A student guesses at each question. Assuming the choices are all equally likely, what is the probability that the student gets all three correct?

Accepted Answer

The probability of guessing each true-false question correctly is 1/2, and the probability of guessing the multiple-choice question correctly is 1/4. Multiplying these probabilities together gives (1/2) * (1/2) * (1/4) = 1/16.

Question 2

Four students' names, including yours, are written on separate slips of paper and placed in a box. The teacher randomly draws two names without replacement. What is the probability that the paper with your name on it will be the second one drawn?

Accepted Answer

The probability of your name being drawn second is the same as the probability of it being drawn first, which is 1 out of 4, since there are four slips of paper and each has an equal chance of being selected at any draw.

Question 3

A medical treatment has a success rate of 0.8. Two patients will be treated with this treatment. Assuming the results are independent for the two patients, what is the probability that neither one of them will be successfully cured?

Accepted Answer

The probability of success for each patient is 0.8, so the probability of failure (not being cured) is 1 - 0.8 = 0.2. Since the results are independent, the probability that neither patient is cured is 0.2 * 0.2 = 0.04.

Question 4

Elizabeth has just put 4 new spark plugs in her car. For each spark plug, the probability that it will fail in the next 50,000 miles is 1/100 (which is 0.01), and is independent from one spark plug to the next. What is the probability that none of the spark plugs will fail in the next 50,000 miles?

Accepted Answer

The probability that one spark plug will not fail is 1 - 0.01 = 0.99. Since the failure of each spark plug is independent, the probability that none of the four spark plugs will fail is (0.99)(0.99)(0.99)(0.99).

Question 5

Tree diagrams are most useful for finding probabilities when

Accepted Answer

Tree diagrams are useful for visualizing and calculating conditional probabilities, especially when you know the probability of one event given another and need to find the reverse conditional probability.

Question 6

Tickets for an upcoming concert are sold out but a local charity is having a raffle and the prize is a pair of tickets to the concert. One hundred people enter the raffle, so each individual who entered has 1/100 probability of winning the pair of tickets. You and a friend both entered. What is the probability that one or the other of you wins the tickets?

Accepted Answer

The probability of you or your friend winning is the sum of your individual probabilities, since these are independent events. Each of you has a 1/100 chance, so together it's 2/100 or 1/50.

Question 7

Use the following information for questions:
 Thirty percent of the students in a high school face a disciplinary action of some kind before they graduate. Of those students, 40% go on to college. Of the 70% who do not face a disciplinary action, 60% go on to college.
-What percent of the students from the high school go on to college?

Accepted Answer

54% of the students from the high school go on to college. This is calculated by taking 30% of students who face disciplinary action and multiplying by 40% (12%) and adding this to 70% of students who do not face disciplinary action multiplied by 60% (42%), resulting in a total of 54%.

Question 8

Use the following information for questions:
 Thirty percent of the students in a high school face a disciplinary action of some kind before they graduate. Of those students, 40% go on to college. Of the 70% who do not face a disciplinary action, 60% go on to college.
-What is the probability that a randomly selected student both faced a disciplinary action and went on to college?

Accepted Answer

The probability that a student faced a disciplinary action and went on to college is found by multiplying the probability of facing a disciplinary action (30% or 0.30) by the probability of going to college given that they faced a disciplinary action (40% or 0.40). Thus, 0.30 * 0.40 = 0.12.

Question 9

Use the following information for questions:
 Thirty percent of the students in a high school face a disciplinary action of some kind before they graduate. Of those students, 40% go on to college. Of the 70% who do not face a disciplinary action, 60% go on to college.
-Given that a randomly selected student goes on to college, what is the probability that he or she faced a disciplinary action?

Accepted Answer

Using Bayes' theorem, the probability that a student faced disciplinary action given they go to college is calculated as (0.3 * 0.4) / [(0.3 * 0.4) + (0.7 * 0.6)] = 0.12 / 0.54 = 0.22.

Question 10

Which of the following is not necessarily true for independent events A and B?

Accepted Answer

For independent events A and B, it's true that P(A and B) = P(A)P(B), P(A|B) = P(A), and P(B|A) = P(B). However, the formula for P(A or B) = P(A) + P(B) - P(A and B) is correct, not P(A or B) = P(A) + P(B) as stated in option D. This is because the intersection of A and B (the events happening together) needs to be subtracted to avoid double counting when A and B are not mutually exclusive.

Question 11

If events A and B are mutually exclusive (disjoint) then

Accepted Answer

Mutually exclusive events cannot occur at the same time, which means the occurrence of one event makes it impossible for the other to occur. This contradicts the definition of independent events, where the occurrence of one event does not affect the probability of the other event occurring. Therefore, mutually exclusive events cannot also be independent.

Question 12

A new drug is successful 80% of the time. A student would like to simulate the probability that if 5 people are given the drug it will be successful on exactly 3 of them. She will randomly choose digits with equal probability from 0, 1, 2, ..., 9 to represent each individual. How should she assign outcomes?

Accepted Answer

Since the drug is successful 80% of the time, the simulation should reflect this probability. Option C, where digits 0 to 7 (8 out of 10 digits, or 80%) represent a successful treatment and digits 8 and 9 (2 out of 10 digits, or 20%) represent failure, accurately simulates this probability.

Question 13

Use the following information for questions:
 In the Banana Republic there are two producers of PCs, referred to as Pac and Bell. Assume that there are no PC imports. The market shares for these two producers are 70% for Pac and 30% for Bell. One executive at Bell proposes a longer warranty period (similar to one given by auto makers) to be offered at a slight extra cost as a plan to increase market share. A market research company appointed by Bell conducts a census of PC owners on their opinion of this warranty proposal. Among owners of a PC made by Pac, 50% like the proposal, 30% are indifferent to it, while the remaining owners oppose it. Among owners of a PC made by Bell, 70% like the proposal, 20% are indifferent to it, and the remaining owners oppose it.
-A PC owner will be selected at random. What is the probability that the person will own a PC made by Bell?

Accepted Answer

The answer of Use the following information for questions:
 In...

Question 14

Use the following information for questions:
 In the Banana Republic there are two producers of PCs, referred to as Pac and Bell. Assume that there are no PC imports. The market shares for these two producers are 70% for Pac and 30% for Bell. One executive at Bell proposes a longer warranty period (similar to one given by auto makers) to be offered at a slight extra cost as a plan to increase market share. A market research company appointed by Bell conducts a census of PC owners on their opinion of this warranty proposal. Among owners of a PC made by Pac, 50% like the proposal, 30% are indifferent to it, while the remaining owners oppose it. Among owners of a PC made by Bell, 70% like the proposal, 20% are indifferent to it, and the remaining owners oppose it.
-A PC owner will be selected at random. What is the probability that the owner will be opposed to the proposal of a new warranty at extra cost?

Accepted Answer

The answer of Use the following information for questions:
 In...

Question 15

Use the following information for questions:
 In the Banana Republic there are two producers of PCs, referred to as Pac and Bell. Assume that there are no PC imports. The market shares for these two producers are 70% for Pac and 30% for Bell. One executive at Bell proposes a longer warranty period (similar to one given by auto makers) to be offered at a slight extra cost as a plan to increase market share. A market research company appointed by Bell conducts a census of PC owners on their opinion of this warranty proposal. Among owners of a PC made by Pac, 50% like the proposal, 30% are indifferent to it, while the remaining owners oppose it. Among owners of a PC made by Bell, 70% like the proposal, 20% are indifferent to it, and the remaining owners oppose it.
-Suppose the selected PC owner is opposed to the new proposal. What is the conditional probability that the person is an owner of a PC made by Bell?

Accepted Answer

The answer of Use the following information for questions:
 In...

Question 16

Use the following information for questions:
 Of the residents of the U.S., 43% live in Northeast/Midwest region while the remaining 57% live in the South/West region. Of the residents in the Northeast/Midwest region, 14% are senior citizens, while the corresponding value for the South/West region is 12%. A citizen will be selected at random.
-What is the probability that the person will be a senior citizen?

Accepted Answer

The answer of Use the following information for questions:
 Of...

Question 17

Use the following information for questions:
 Of the residents of the U.S., 43% live in Northeast/Midwest region while the remaining 57% live in the South/West region. Of the residents in the Northeast/Midwest region, 14% are senior citizens, while the corresponding value for the South/West region is 12%. A citizen will be selected at random.
-What is the probability that the person will be a senior citizen from the Northeast/Midwest region?

Accepted Answer

The answer of Use the following information for questions:
 Of...

Question 18

Use the following information for questions:
 Of the residents of the U.S., 43% live in Northeast/Midwest region while the remaining 57% live in the South/West region. Of the residents in the Northeast/Midwest region, 14% are senior citizens, while the corresponding value for the South/West region is 12%. A citizen will be selected at random.
-What is the probability that the person will be from the Northeast/Midwest region, given that he/she is a senior citizen?

Accepted Answer

The answer of Use the following information for questions:
 Of...

Question 19

The statistics of a particular basketball player state that he makes, on average, 4 out of every 5 free throw attempts. This player is about to make 5 free throw attempts. We wish to calculate the probability that all 5 attempts result in a goal. Explain how you would choose random digits to simulate this scenario.

Accepted Answer

The answer of The statistics of a particular basketball player...

Question 20

A standard deck of cards has 52 cards. The cards have one of 4 suites: 13 cards are hearts, 13 cards are diamonds, 13 cards are clubs, and 13 cards are spades. We will draw cards from a shuffled deck of cards with replacement (after we draw each card, we will record the suite and put it back in the deck). We will keep going until we draw hearts. We wish to calculate the probability of having to draw 3 or fewer cards. Explain how you would choose random digits to simulate this scenario.

Accepted Answer

The answer of A standard deck of cards has 52...

Quiz 7: Probability

A short quiz has two true-false questions and one multiple-choice question with four choices. A student guesses at each question. Assuming the choices are all equally likely, what is the probability that the student gets all three correct?

Four students' names, including yours, are written on separate slips of paper and placed in a box. The teacher randomly draws two names without replacement. What is the probability that the paper with your name on it will be the second one drawn?

A medical treatment has a success rate of 0.8. Two patients will be treated with this treatment. Assuming the results are independent for the two patients, what is the probability that neither one of them will be successfully cured?

Elizabeth has just put 4 new spark plugs in her car. For each spark plug, the probability that it will fail in the next 50,000 miles is 1/100 (which is 0.01) , and is independent from one spark plug to the next. What is the probability that none of the spark plugs will fail in the next 50,000 miles?

Tree diagrams are most useful for finding probabilities when

Which of the following is not necessarily true for independent events A and B?

If events A and B are mutually exclusive (disjoint) then