Section 7.5 Conditional Probability and Independent Eventsmayaj/Chapter7_Sec7.5_7... · the products, the second machine 25%, and the third machine 40%. The ﬁrst machine produces

Section 7.5 Conditional Probability and Independent Events

Conditional Probability of an Event

If A and B are events in an experiment and P (A) 6= 0, then the conditional probability that the event

B will occur given that the event A has already occurred is

P (B|A) = P (A \ B)

P (A)

1. A pair of fair 6-sided dice is rolled. What is the probability that a 2 is rolled if it is known that

the sum of the numbers landing uppermost is less than or equal to 7? (Give answers as an exact

fraction.)

2. A company surveyed 1000 people on their age and the number of jeans purchased annually. The

results of the poll are shown in the table.

0 1 2 3 or More Total

U¯nder 12 0 70 76 64 210

1¯2-18 17 54 154 55 280

1¯9-25 39 57 137 51 280

o¯ver 25 59 81 69 21 230

T¯otal 115 262 432 191 1000

A person is selected at random. Use the table to answer these questions. Round your answers to

three decimal places.

(a) What is the probability that the person, who is over 18, purchases 2 pairs of jeans annually?

(b) What is the probability that a person who purchased less than 3 pairs of jeans each year will

be in the age group 12-18?

Product Rule

P (A \ B) = P (A) · P (B|A)

3. From the tree diagram find the following.

(a) P (A \ E)

(b) P (A)

(c) P (A|E)

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4. In a survey of 1000 eligible voters selected at random, it was found that 200 had a college degree.

Additionally, it was found that 70% of those who had a college degree voted in the last presidential

election, whereas 45% of the people who did not have a college degree voted in the last presidential

election. Assuming that the poll is representative of all eligible voters, find the probability that

an eligible voter selected at random will have the following characteristics. (Round answers to

three decimal places.)

(a) The voter had a college degree and voted in the last presidential election.

(b) The voter did not have a college degree and did not vote in the last presidential election.

(c) The voter voted in the last presidential election.

(d) The voter did not vote in the last presidential election.


5. Two machines turn out all the products in a factory, with the first machine producing 75% of the

product and the second 25%. The first machine produces defective products 5% of the time and

the second machine 7% of the time.

(a) What is the probability that a defective part is produced at this factory given that it was

made on the first machine?

(b) What is the probability that a defective part is produced at this factory?

Independent Events If A and B are independent events, then

P (A|B) = P (A) and P (B|A) = P (B)

Test for the Independence of Two Events Two events A and B are independent if and only

if

P (A \B) = P (A) · P (B)


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6. The personnel department of Franklin National Life Insurance compiled the accompanying data

regarding the income and education of its employees.

Income 60,000 or Below Income Above 60,000

Noncollege Graduate 2050 830

College Graduate 380 740

Let A be the event that a randomly chosen employee has a college degree, and let B be the event

that the chosen employee’s income is more than $60, 000.

(a) Find each of the following probabilities. (Round answers to four decimal places.)

P (A)

P (B)

P (A \B)

P (B|A)

P (B|Ac)

(b) Are the events A and B independent events?

7. Suppose A and B are two events of a sample space S where P (A) = 0.28, P (B) = 0.24, and

P (A [ B) = 0.42.

(a) What is P (A \ B)?

(b) Are A and B independent events?


8. An experiment consists of two independent trials. The outcomes of the first trial are A, B, and

C, with probabilities of occurring equal to 0.2, 0.2, and 0.6, respectively. The outcomes of the

second trial are E and F , with probabilities of occurring equal to 0.3 and 0.7. Draw a tree diagram

representing this experiment. Use this tree diagram to find the probabilities below.

(a) P (B)

(b) P (F |B)

(c) P (B \ F )

(d) P (F )

(e) Does P (B \ F ) = P (B) · P (F )

(f) Are B and F independent events?

9. Dystopia county has three bridges. In the next year, the Elder bridge has an 8% chance of

collapse, the Younger bridge has a 3% chance of collapse, and the Ancient bridge has a 19%

chance of collapse. What is the probability that exactly one of these bridges will collapse in the

next year? (Round answer to four decimal places)


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10. If A and B are independent events, P (A) = 0.35, and P (B) = 0.55, find the probabilities below.

(Enter answers to four decimal places.)

(a) P (A \ B)

(b) P (A [ B)

(c) P (A|B)

(d) P (Ac [ Bc)


Section 7.6 Bayes’ Theorem

Bayes’ Theorem

Let A1, A2, · · · , An be a partition of a sample space S, and let E be an event of the experiment such

that P (E) 6= 0 and P (Ai) 6= 0 for 1 i n. Then the conditional probability P (Ai|E) (1 i n) is

given by

P (Ai|E) =P (Ai) · P (E|Ai)

P (A1) · P (E|A1) + P (A2) · P (E|A2) + · · ·+ P (An) · P (E|An)

Recall from section 7.5 that P (Ai \ E) = P (Ai) · P (E|Ai). Also, P (E) = P (A1) · P (E|A1) + P (A2) ·P (E|A2)+ · · ·+P (An) ·P (E|An). Therefore, we could use the conditional probability rule from section

7.5 and say that

P (Ai|E) =P (Ai \ E)

P (E)

1. Find P (F |B) and P (E|A) using the tree diagram. (Round answers to three decimal places.)


2. A survey involving 700 likely Democratic voters and 200 likely Republican voters asked the ques-

tion: Do you support or oppose legislation that would require trigger locks on guns, to prevent

misuse by children? The following results were obtained:

Answer Democrats, % Republicans, %

Support 85 73

Oppose 8 17

Don’t know/refused 7 10

If a randomly chosen respondent in the survey answered ”support,” what is the probability that

he or she is a likely Republican voter? (Round answer to three decimal places.)

3. Applicants who wish to be admitted to a certain professional school in a large university are

required to take a screening test devised by an educational testing service. From past results, the

testing service has established that 70% of all applicants are eligible for admission and that 90%

of those who are eligible for admission pass the exam, whereas 14% of those who are ineligible for

admission pass the exam. (Round answers to three decimal places.)


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(a) What is the probability that an applicant for admission passed the exam?

(b) What is the probability that an applicant for admission who passed the exam was actually

ineligible?

4. There are three jars that each contain 10 marbles. The first contains 3 white marbles and 7 red

marbles, the second 6 white and 4 red, and the third all 10 white. An experiment consists of first

selecting a jar at random. (Assume each jar has an equal probability of being selected.) After a

jar is selected, a marble is randomly drawn from this jar, noting its color. If the marble drawn

was white, find the probability that the third jar was selected. (Round answer to three decimal

places.)


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5. The O�ce of Admissions and Records of a large western university released the accompanying

information concerning the contemplated majors of its freshman class. (Round answers to three

decimal places.)

% of Freshmen % of Major % of Major

Choosing That is That is

Major This Major Female Male

Business 20 36 64

Humanities 9 65 35

Education 10 65 35

Social science 10 52 48

Natural sciences 8 57 43

Other 43 47 53

(a) What is the probability that a student selected at random from the freshman class is a

female?

(b) What is the probability that a business student selected at random from the freshman class

is a male?

(c) What is the probability that a female student selected at random from the freshman class is

majoring in business?


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6. Three machines turn out all the products in a factory, with the first machine producing 35% of

the products, the second machine 25%, and the third machine 40%. The first machine produces

defective products 6% of the time, the second machine 17% of the time and the third machine 4%

of the time. What is the probability that a non-defective product came from the second machine?

(Round answer to four decimal places.)

7. Box A contains seven white marbles and five black marbles. Box B contains six white marbles

and four black marbles. An experiment consists of first selecting a marble at random from Box

A. The marble is transferred to Box B and then a second marble is drawn from Box B. What is

the probability that the first marble was white given that the second marble was white? (Round

answer to three decimal places.)


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8. A medical test has been designed to detect the presence of a certain disease. Among people who

have the disease, the probability that the disease will be detected by the test is 0.91. However,

among those who do not have the disease, the probability that the test will detect the presence

of the disease is 0.04. It is estimated that 3% of the population who take this test actually have

the disease. (Round answers to three decimal places.)

(a) If the test administered to an individual is positive (the disease is detected), what is the

probability that the person actually has the disease?

(b) If the test administered to an individual is negative (the disease is not detected), what is the

probability that the person actually does have the disease?


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Section 7.5 Conditional Probability and Independent Eventsmayaj/Chapter7_Sec7.5_7... · the products, the second machine 25%, and the third machine 40%. The ﬁrst machine produces