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.

(a) 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.)

(b) If at least one 3 is rolled, what is the probability that the sum is 6? (Give answers as an

exact fraction.)

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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)

2 Fall 2019, Maya Johnson

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Tree Diagram: A tree diagram is a way to visually keep track of a multi-step experiment.

Each set of “branches” of the tree diagram act like probability distributions with the same prop-

erties of a probability distribution.

Analyzing a Tree Diagram:

3. From the tree diagram find the following.

(a) P (C \ F )

(b) P (B)

(c) P (A|E)

3 Fall 2019, Maya Johnson

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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 voted 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. What is the probability that a defective part is produced at

this factory given that it was made on the first machine?

4 Fall 2019, Maya Johnson

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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)

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)

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

5 Fall 2019, Maya Johnson

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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. 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 (Ac \ Bc)

(c) P (A|B)

(d) P (Ac [ Bc)

6 Fall 2019, Maya Johnson

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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)

7 Fall 2019, Maya Johnson

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