Question 1

If a dime is tossed four times and comes up tails all four times, the probability of heads on the fifth trial is
A)smaller than the probability of tails
B)larger than the probability of tails
C)1/2
D)1/32

Accepted Answer

The probability of heads on the fifth trial is independent of the outcomes of the previous four trials. Each trial has a 50/50 chance of heads or tails, so the probability of heads on the fifth trial is also 1/2, regardless of the previous outcomes.

Question 2

If a six sided die is tossed two times and "3" shows up both times, the probability of "3" on the third trial is
A)much larger than any other outcome
B)much smaller than any other outcome
C)1/6
D)1/216

Accepted Answer

The outcome of each roll of the die is independent of the previous rolls, so the probability of rolling a "3" on the third trial is still 1/6, the same as any other outcome. The previous rolls do not affect the probability of future rolls.

Question 3

An experiment consists of three steps. There are four possible results on the first step, three possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is
A)9
B)14
C)24
D)36

Accepted Answer

The total number of experimental outcomes is found by multiplying the number of possible outcomes at each step. Thus, the total number of outcomes is:
4 x 3 x 2 = 24. Therefore, the correct choice is C, 24.

Question 4

Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there?
A)20
B)7
C)5!
D)10

Accepted Answer

There are 5 options for the first letter and 4 options left for the second letter. However, since AB and BA are the same selection, we must divide by 2 to avoid double counting. 
Total selections = 5 x 4 / 2 = 10.

Question 5

Each customer entering a department store will either buy or not buy some merchandise. An experiment consists of following 3 customers and determining whether or not they purchase any merchandise. The number of sample points in this experiment is
A)2
B)4
C)6
D)8

Accepted Answer

The answer of Each customer entering a department store will...

Question 6

Assuming that each of the 52 cards in an ordinary deck has a probability of 1/52 of being drawn, what is the probability of drawing a black ace?
A)1/52
B)2/52
C)3/52
D)4/52

Accepted Answer

The answer of Assuming that each of the 52 cards...

Question 7

Two events, A and B, are mutually exclusive and each have a nonzero probability. If event A is known to occur, the probability of the occurrence of event B is
A)one
B)any positive value
C)zero
D)any value between 0 to 1

Accepted Answer

If two events are mutually exclusive, it means that they cannot occur at the same time. If event A has occurred, this means that event B cannot occur. Therefore, the probability of the occurrence of event B when event A has occurred is zero.

Question 8

If P(A) = 0.4, P(B | A) = 0.35, P(A $\cup$ B) = 0.69, then P(B) =
A)0.14
B)0.43
C)0.75
D)0.59

Accepted Answer

Using the law of total probability: P(A F1F1F1S1F1F1F10 B) = P(B | A) * P(A) + P(B | A^c) * P(A^c) Where A^c represents the complement of event A. We can rearrange and solve for P(B): P(B) = [P(A F1F1F1S1F1F1F10 B) - P(B | A^c) * (1 - P(A))] / P(A) Substituting the given values: P(B) = [0.69 - P(B | A^c) * 0.6] / 0.4 We don't have enough information to directly solve for P(B | A^c), but we can use the fact that P(B) must be between 0 and 1: 0 < P(B) < 1 0 < [0.69 - P(B | A^c) * 0.6] / 0.4 < 1 0 < 1.725 - 1.5P(B | A^c) < 2.5 -1.725 < -1.5P(B | A^c) < 0.775 1.15 > P(B | A^c) > -0.517 Since probabilities must be between 0 and 1, we can conclude that: 0.35 < P(B) < 1 Eliminating the impossible lower bound, we get: 0.43 < P(B) < 1 The only answer choice within this range is B, 0.43.

Question 9

The set of all possible outcomes of an experiment is
A)an experiment
B)an event
C)the population
D)the sample space

Accepted Answer

The set of all possible outcomes of an experiment is called the sample space. Therefore, the best choice is D, as it directly relates to the definition given in the question. Choices A and B are incorrect as they do not accurately reflect the definition of the sample space, and choice C is incorrect as the population refers to a specific group being studied, not the set of all possible outcomes of an experiment.

Question 10

Three applications for admission to a local university are checked, and it is determined whether each applicant is male or female. The number of sample points in this experiment is
A)2
B)4
C)6
D)8

Accepted Answer

The answer of Three applications for admission to a local...

Question 11

Given that event E has a probability of 0.31, the probability of the complement of event E
A)cannot be determined with the above information
B)can have any value between zero and one
C)0.69
D)is 0.31

Accepted Answer

The probability of the complement of event E is equal to 1 minus the probability of event E. Therefore, the probability of the complement of event E is 1 - 0.31 = 0.69. Thus, the correct answer is C.

Question 12

Since the sun must rise tomorrow, then the probability of the sun rising tomorrow is
A)much larger than one
B)zero
C)infinity
D)None of these alternatives is correct.

Accepted Answer

The answer of Since the sun must rise tomorrow, then...

Question 13

An experiment consists of tossing 4 coins successively. The number of sample points in this experiment is
A)16
B)8
C)4
D)2

Accepted Answer

For each coin toss, there are 2 possible outcomes (heads or tails). Therefore, for 4 coin tosses, we have 2x2x2x2 = 16 possible outcomes (sample points).

Question 14

If A and B are independent events with P(A) = 0.65 and P(A $\cap$ B) = 0.26, then, P(B) =
A)0.400
B)0.169
C)0.390
D)0.650

Accepted Answer

The answer of If A and B are independent events...

Question 15

Assume your favorite football team has 2 games left to finish the season. The outcome of each game can be win, lose or tie. The number of possible outcomes is
A)2
B)4
C)6
D)9

Accepted Answer

There are 3 possible outcomes for each game (win, lose, or tie) and there are 2 games, so the total number of possible outcomes is 3x3=9. Therefore, the best choice is D.

Question 16

Events A and B are mutually exclusive if their joint probability is
A)larger than 1
B)less than zero
C)zero
D)infinity

Accepted Answer

Mutually exclusive events cannot occur simultaneously, therefore their joint probability is zero.

Question 17

Initial estimates of the probabilities of events are known as
A)sets
B)posterior probabilities
C)conditional probabilities
D)prior probabilities

Accepted Answer

Initial estimates of the probabilities of events are known as prior probabilities.

Question 18

The sum of the probabilities of two complementary events is
A)Zero
B)0.5
C)0.57
D)1.0

Accepted Answer

Complementary events are two events that cover all possible outcomes and have no overlap. The sum of their probabilities should equal 1. For example, if event A is the probability of getting heads on a coin flip and event B is the probability of getting tails on a coin flip, then event A and B are complementary events. The sum of their probabilities would be 0.5 + 0.5 = 1. Therefore, the correct answer is D, 1.0.

Question 19

The addition law is potentially helpful when we are interested in computing the probability of
A)independent events
B)the intersection of two events
C)the union of two events
D)conditional events

Accepted Answer

The addition law is used to calculate the probability of the union of two events - the probability that at least one of the events occurs.

Question 20

Two events with nonzero probabilities
A)can be both mutually exclusive and independent
B)can not be both mutually exclusive and independent
C)are always mutually exclusive
D)are always independent

Accepted Answer

If two events have nonzero probabilities, they can not be both mutually exclusive and independent. Mutually exclusive events can not occur simultaneously, while independent events have no effect on each other. Therefore, if they can not occur together, they can not be independent.

Quiz 4: Introduction to Probability

If a dime is tossed four times and comes up tails all four times, the probability of heads on the fifth trial is

If a six sided die is tossed two times and "3" shows up both times, the probability of "3" on the third trial is

An experiment consists of three steps. There are four possible results on the first step, three possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is

Of five letters (A, B, C, D, and E) , two letters are to be selected at random. How many possible selections are there?

Each customer entering a department store will either buy or not buy some merchandise. An experiment consists of following 3 customers and determining whether or not they purchase any merchandise. The number of sample points in this experiment is

Assuming that each of the 52 cards in an ordinary deck has a probability of 1/52 of being drawn, what is the probability of drawing a black ace?

Two events, A and B, are mutually exclusive and each have a nonzero probability. If event A is known to occur, the probability of the occurrence of event B is

If P(A) = 0.4, P(B | A) = 0.35, P(A $\cup$ B) = 0.69, then P(B) =

The set of all possible outcomes of an experiment is

Three applications for admission to a local university are checked, and it is determined whether each applicant is male or female. The number of sample points in this experiment is

Given that event E has a probability of 0.31, the probability of the complement of event E

Since the sun must rise tomorrow, then the probability of the sun rising tomorrow is

An experiment consists of tossing 4 coins successively. The number of sample points in this experiment is

If A and B are independent events with P(A) = 0.65 and P(A $\cap$ B) = 0.26, then, P(B) =

Assume your favorite football team has 2 games left to finish the season. The outcome of each game can be win, lose or tie. The number of possible outcomes is

Events A and B are mutually exclusive if their joint probability is

Initial estimates of the probabilities of events are known as

The sum of the probabilities of two complementary events is

The addition law is potentially helpful when we are interested in computing the probability of

Two events with nonzero probabilities