Deck 6: Probability

Question 1

If you roll a fair (unbiased) die 60 times, you should expect an odd number to appear:

A)in each of the first 30 rolls.
B)30 out of the 60 rolls.
C)on every other roll.
D)40 out of the 60 rolls.

Answer 1

30 out of the 60 rolls.

Question 2

Which of the following statements is always correct?

A)P(A $\cap$ B) = P(A) * P(B).
B)P(A $\cup$ B) = P(A) + P(B).
C)P(A $\cup$ B) = P(A) + P(B) + P(A $\cap$ B).
D)P(A) = 1 - P( $\bar { A }$ ).

Answer 2

P(A) = 1 - P( $\bar { A }$ ).

Question 3

A useful graphical method of constructing the sample space for an experiment is:

A)a tree diagram.
B)a pie chart.
C)a histogram.
D)an ogive.

Answer 3

a tree diagram.

Question 4

If A and B are mutually exclusive events with P(A) = 0.80, then P(B):

A)Can take any value between 0 and 1.
B)Can take any value between 0 and 0.80.
C)Can be larger than 0.80.
D)Cannot be larger than 0.20.

Question 5

If A and B are mutually exclusive events, with P(A) = 0.20 and P(B) = 0.30, then P(A $\cap$ B) is:

A)0.50.
B)0.10.
C)0.00.
D)0.06.

Question 6

If P(A) = 0.20, P(B) = 0.30 and P(A $\cap$ B) = 0.00, then A and B are:

A)dependent events.
B)independent events.
C)mutually exclusive events.
D)complementary events.

Question 7

When a fair die is rolled once, the sample space consists of the following six outcomes: 1, 2, 3, 4, 5, 6. Given this sample space, which of the following is a simple event?

A)Even number.
B)More than 4.
C)Less than 1.
D)4.

Question 8

Suppose P(A) = 0.25. The probability of complement of A is:

A)0.25.
B)0.50.
C)−0.25.
D)0.75.

Question 9

If P(A) = 0.65, P(B) =0.76 and P(A $\cap$ B) =0.80, then P(A $\cap$ B) is:

A)0.65.
B)0.61.
C)0.80.
D)0.02.

Question 10

If P(A) = 0.60, P(B) = 0.58, and P(A $\cup$ B) = 0.70, then P(A $\cap$ B) is:

A)0.60.
B)0.70.
C)0.48.
D)0.58.

Question 11

Which of the following is not an approach to assigning probabilities?

A)Classical approach
B)Trial-and-error approach
C)Relative frequency approach
D)Subjective approach

Question 12

Two events A and B are said to mutually exclusive if:

A)P(A | B) = 1.
B)P(B | A) =1.
C)P(A $\cap$ B) =1.
D)P(A $\cap$ B) = 0.

Question 13

If A and B are independent events, with P(A) = 0.50 and P(B) = 0.70, then the probability that A occurs or B occurs or both occur is:

A)1.20.
B)0.20.
C)0.85.
D)0.10.

Question 14

Two events A and B are said to be independent if:

A)P(A $\cap$ B) = P(A) * P(B).
B)P(B | A) = P(B).
C)P(A | B) = P(A).
D)All of these choices are correct.

Question 15

If P(A) = 0.35, P(B) = 0.45 and P(A $\cap$ B) =0.20, then P(A | B) is:

A)0.80.
B)0.60.
C)0.44.
D)0.57.

Question 16

Which of the following is a requirement of the probabilities assigned to the outcomes $O _ { i }$ ?

A)P(Oi) $\le$ 0.
B)P(Oi) $\ge$ 0.
C)0 $\le$ P(Oi) $\le$ 0, for each i.
D)P(Oi) = 1 + P( $\bar { O } _ { i }$ ).

Question 17

The collection of all possible outcomes of an experiment is called:

A)a simple event.
B)a sample space.
C)a sample.
D)a population.

Question 18

If the events A and B are independent, with P(A) = 0.30 and P(B) = 0.40, then the probability that both events will occur simultaneously is:

A)0.10.
B)0.12.
C)0.70.
D)0.75.

Question 19

An approach of assigning probabilities that assumes that all outcomes of the experiment are equally likely is referred to as the:

A)subjective approach.
B)objective approach.
C)classical approach.
D)relative frequency approach.

Question 20

If A and B are independent events with P(A) = 0.60 and P(A/B) = 0.60, then P(B) is:

A)1.20.
B)0.60.
C)0.36.
D)P(B) cannot be determined with the information given.

Question 21

The annual estimate of the number of deaths of infants is an example of the classical approach to probability.

Question 22

If a coin is tossed three times, and a statistician predicts that the probability of obtaining three heads in a row is 0.125, which of the following assumptions is irrelevant to his prediction?

A)The events are dependent.
B)The events are independent.
C)The coin is unbiased.
D)All of the above assumptions are relevant to his prediction.

Question 23

If an experiment consists of five outcomes, with $P \left( Q _ { 1 } \right) =$ 0.10, $P \left( O _ { 2 } \right) =$ 0.10, $P \left( O _ { 3 } \right) =$ 0.30, $P \left( O _ { 4 } \right) =$ 0.25, then $P \left( O _ { 5 } \right)$ is:

A)0.75.
B)0.25.
C)0.20.
D)0.80.

Question 24

Two or more events are said to be independent when the occurrence of one event has an effect on the probability that another will occur.

Question 25

If the event of interest is A, the probability that A will not occur is the complement ofA.

Question 26

When events are mutually exclusive, they can happen at the same time.

Question 27

The relative frequency approach to probability depends on the law of large numbers.

Question 28

If A and B are independent events, with P(A) = 0.20 and P(B) =0.60, then P(A | B) is:

A)0.20.
B)0.60.
C)0.40.
D)0.80.

Question 29

Probability refers to a number between 0 and 1 (inclusive), which expresses the chance that an event will occur.

Question 30

Conditional probability is the probability that an event will occur, given that another event will also occur.

Question 31

Of the last 400 customers entering a supermarket, 20 have purchased a mobile phone. If the classical approach for assigning probabilities is used, the probability that the next customer will purchase a mobile phone is:

A)0.80.
B)0.20.
C)0.05.
D)0.50.

Question 32

Five students from a statistics class have formed a study group. Each may or may not attend a study session. Assuming that the members will be making independent decisions on whether or not to attend, there are 32 different possibilities for the composition of the study session.

Question 33

The probability of event A and event B occurring must be equal to 1.

Question 34

If event A does not occur, then its complement $\bar { A }$ must occur.

Question 35

Marginal probability is the probability that a given event will occur, with no other events taken into consideration.

Question 36

If A and B are mutually exclusive events, with P(A) = 0.30 and P(B) = 0.40, then P(A $\cup$ B) is:

A)0.10.
B)0.12.
C)0.70.
D)None of these choices are correct.

Question 37

Based on past exam results in principles of accounting you estimate that there is an 83% chance of passing the exam. This is an example of the subjective approach to probability.

Question 38

If we wished to determine the probability that one or more of several events will occur in an experiment, we would use addition rules.

Question 39

If P(A) = 0.25 and P(B) = 0.65, then P(A $\cap$ B) is:

A)0.25.
B)0.40.
C)0.90.
D)P(A $\cap$ B) cannot be determined from the information given.

Question 40

The classical approach to assigning probability can be applied for experiments that have equally likely outcomes.

Question 41

Assume that A and B are independent events, with P(A) = 0.30 and P(B) = 0.50. The probability that both events will occur simultaneously is 0.80.

Question 42

Suppose P(A) = 0.10, P(B) = 0.70, and P(B/A) = 0.80.
a. Find P(A $\cap$ B).
b. Find P(A $\cup$ B).
c. Find P(A | B).

Question 43

An experiment consists of tossing three fair (unbiased) coins simultaneously. This experiment has eight possible outcomes.

Question 44

At the beginning of each year, an investment newsletter predicts whether or not the stock market will rise over the coming year. Historical evidence reveals that there is a 75% chance that the stock market will rise in any given year. The newsletter has predicted a rise for 80% of the years when the market actually rose, and has predicted a rise for 40% of the years when the market fell. Find the probability that the newsletter's prediction for next year will be correct.

Question 45

According to an old song lyric, 'love and marriage go together like a horse and carriage'. Let love be event A and marriage be event
B.Events A and B cannot be mutually exclusive.

Question 46

The probability of the union of two mutually exclusive events A and B is P(A $\cup$ B) = P(A) + P(B).

Question 47

Bayes' Law is a method of revising probabilities after another event has occurred.

Question 48

Three candidates for the presidency of a university's student union, Alice, Brenda and Cameron, are to address a student forum. The forum's organiser is to select the order in which the candidates will give their speeches, and must do so in such a way that each possible order is equally likely to be selected.
a. What is the random experiment?
b. List the simple events in the sample space.
c. Assign probabilities to the simple events.
d. What is the probability that Cameron will speak first?
e. What is the probability that one of the women will speak first?
f. What is the probability that Alice will speak before Cameron does?

Question 49

A PhD graduate has applied for a job with two universities, A and B.The graduate feels that she has a 60% chance of receiving an offer from university A, and a 30% chance of receiving an offer from university B.If she receives an offer from university B, she believes that she has an 70% chance of receiving an offer from university A. a.What is the probability that both universities will make her an offer?
b.What is the probability that at least one university will make her an offer?
c.If she receives an offer from university B, what is the probability that she will not receive an offer from university A?

Question 50

If events A and B have nonzero probabilities, then they can be both independent and mutually exclusive.

Question 51

Suppose A and B are two independent events, with P(A) = 0.20 and P(B) = 0.60.
a. Find P(B | A).
b. Find P(A | B).
c. Find P(A and B).
d. Find P(A or B).

Question 52

If A and B are independent events, with P(A) = 0.30 and P(B) = 0.50, then P(B | A) is 0.10.

Question 53

Two events A and B are said to be independent if P(A $\cap$ B) = P(A)+ P(B).

Question 54

The relative frequency approach is not useful in interpreting probability statements such as those heard from weather forecasters or scientists.

Question 55

There are three approaches to determining the probability that an outcome will occur: the classical, relative frequency, and subjective approaches. Which is most appropriate in determining the probability of the following outcomes?
a. A flipped coin will land on tails.
b. The probability of your favourite team winning the finals.
c. Five of the next 20 new cars sold in Adelaide will be imported cars.

Question 56

An effective and simple method of applying the probability rules is the probability tree, wherein the events of an experiment are represented by lines.

Question 57

Jim and John go to a coffee shop during their lunch break and toss a coin to see who will pay. The probability that John will pay three days in a row is 0.125.

Question 58

Two events A and B are said to mutually exclusive if P(A) = P(B).

Question 59

Given that events A and B are independent, and that P(A) = 0.9 and P(B | A) = 0.5, then P(A $\cap$ B) = 0.45.

Question 60

When it is not reasonable to use the classical approach to assigning probabilities to the outcomes of an experiment, and there is no history of the outcomes, we have no alternative but to employ the subjective approach.

Question 61

An insurance company has collected the following data on the gender and marital status of 300 customers.
Marital Status $$\begin{array} { | l | c | c | c | } \hline \text { Gender } & \text { Single } & \text { Married } & \text { Divorced } \\\hline \text { Male } & 25 & 125 & 30 \\\hline \text { Female } & 50 & 50 & 20 \\\hline\end{array}$$ Suppose that a customer is selected at random. Find the probability that the customer selected is:
a. a married female.
b. not single.
c. married, if the customer is male.
d. female or divorced.
e. Are gender and marital status mutually exclusive? Explain using probabilities.
f. Is marital status independent of gender? Explain using probabilities.

Question 62

Suppose P(A) = 0.30, P(B) = 0.40, and P(B /A) = 0.60.
a. Find P(A $\cap$ B).
b. Find P(A $\cup$ B).
c. Find P(A /B).

Question 63

Suppose P(A) = 0.50, P(B) = 0.30, and P(A or B) = 0.80.
a. Find $P ( A \cap B )$
b. Find $P ( B \mid A )$
c. Are A and B mutually exclusive events? Explain using probabilities.

Question 64

An financial advisor tells you that in her estimation there is an 85% chance that a particular stock's price will increase over the next three weeks.
a. Which approach was used to produce this figure?
b. Interpret the 85% probability.

Question 65

Find the probability that the ice cream was sold on a cone and the flavour was:
a. chocolate.
b. strawberry.
c. vanilla.

Question 66

A statistics professor classifies his students according to their gender and the number of hours of paid work they do a week. The following table gives the proportions of students falling into the various categories. One student is selected at random.
Paid Work (hours/week) a. If the student selected is female, what is the probability that he works between 1 and 8 hours a week?
b. If the selected student works more than 16 hours a week, what is the probability that the student is male?
c. What is the probability that the student selected is female or does do any paid work or both?
d. Is gender independent of the number of hours of paid work done a week? Explain using probabilities.

Question 67

Suppose $$P ( A ) = 0.1 , P ( B ) = 0.5 , \text { and } P ( A \cap B ) = 0$$
a. Find $P ( A \cup B )$
b. Are A and B independent events? Explain.
c. Are A and B mutually exclusive events? Explain.

Question 68

An insurance company has recently recruited ten graduates, four men and six women. Two of the graduates are to be selected at random to work in the firm's suburban office.
a. What is the probability that two men will be selected?
b. What is the probability that at least one man will be selected?

Question 69

Suppose A and B are two mutually exclusive events for which P(A) = 0.25 and P(B) = 0.60.
a. Find
b. Find
c. Find
d. Are A and B independent events? Explain using probabilities.

Question 70

A woman is expecting her second child. Her doctor has told her that she has a 50-50 chance of having another girl. If she has another girl, there is a 90% chance that she will be taller than the first. If she has a boy, however, there is only a 25% chance that he will be taller than the first child. Find the probability that the woman's second child will be taller than the first.

Question 71

A pharmaceutical firm has discovered a new diagnostic test for a certain disease that has infected 1% of the population. The firm has announced that 95% of those infected will show a positive test result, while 98% of those not infected will show a negative test result. What proportion of test results are correct?

Question 72

The sample space of the toss of a fair die is S = {1, 2, 3, 4, 5, 6}. If the die is balanced, each simple event has the same probability. Find the probability of the following events.
a. Equal to 1.
b. A number greater than 3.
c. A number greater than 6.
d. A number between 2 and 4, inclusive.

Question 73

Is it possible to have two events for which Explain.

Question 74

Find the probability that the ice cream was sold in a cup and the flavour was:
a. chocolate.
b. strawberry.
c. vanilla.

Question 75

A standard admissions test was given at three locations. One thousand students took the test at location A, 600 students at location B, and 400 students at location C.The percentages of students from locations A, B and C who passed the test were 70%, 68% and 77%, respectively.One student is selected at random from among those who took the test.
a.What is the probability that the selected student passed the test?
b.If the selected student passed the test, what is the probability that the student took the test at location B?
c.What is the probability that the selected student took the test at location C and failed?

Question 76

Suppose P(A) = 0.40, P(B) = 0.50, and P(A $\cup$ B) = 0.70.
a. Find P(A $\cap$ B).
b. Find P(B | A).
c. Are A and B independent events? Explain using probabilities.

Question 77

Suppose a. Find
b. Are A and B mutually exclusive events

Question 78

The following table shows the numbers of cars sold by a car dealer during the last 30 weeks. a. Define the random variable of interest to the dealer.
b. List the simple events in the sample space.
c. Assign probabilities to the simple events and show the probability distribution.
d. What approach have you used in determining the probabilities in part (c)?
e. What is the probability of selling no more than four cars in any given week?

Question 79

Suppose P( ) = 0.10, P( | A) = 0.40, and P( | ) = 0.50.
a. Find P(A).
b.
c. Find

Question 80

A survey of a magazine's subscribers indicates that 40% own a home, 80% own a car, and 90% of the homeowners who subscribe also own a car. What proportion of subscribers:
a. own both a car and a house?
b. own a car or a house, or both?
c. own neither a car nor a house?