10.5 Find Probabilities of Independent and Dependent Events

VOCABULARY

Independent events

Two events such that the occurrence of one has no effect on the occurrence of the other

Dependent events

Two events such that the occurrence of one affects the occurrence of the other

Conditional probability

The probability that event B will occur given that event A has occurred is called the conditional probability of B given A.

PROBABILITY OF INDEPENDENT EVENTS

If A and B are independent events, then the probability that both A and B occur is:
P(A and B) = _P(A) · P(B)_

More generally, the probability that n independent events occur is the _product_ of the n probabilities of the individual events.

Example 1

Find probability of three independent events

Attendance Every morning, one student in a class of 24 students is randomly chosen to take attendance. What is the probability that the same student will be chosen three days in a row?

Let events A, B, and C be the student being chosen on the first, second, and third day, respectively. The three events are independent. So, the probability is:

P(A and B and C) = __P(A) · P(B) · P(C)__

= _{___________ » _0.0000723_}

Example 2

Use a complement to find a probability

Manufacturing A manufacturer has found that 2 out of every 500 coffee pots produced are defective. What is the probability that at least one coffee pot is defective in the first 300 coffee pots made?

Solution

The probability of not making a defective coffee pot is:

P(coffee pot is not defective) = _{_______}= __0.996__

Each coffee pot made is an independent event. So, the probability of making at least one defective coffee pot is:

P(coffee pot is defective) = 1 - [P(coffee pot is not defective)]³⁰⁰

= _1- (0.996)³⁰⁰_ » _0.700_

Complete the following exercises.

1. During a high school track meet, each race consists of 9 competitors who are randomly assigned lanes from 1 to 9. What is the probability that a runner will draw lanes 1, 2, or 3 in the three races in which he competes?

0.03

2. A manufacturer has found that 6 out of every 450 batteries produced is defective. What is the probability that at least one battery is defective in the first 100 produced?

0.739

PROBABILITY OF DEPENDENT EVENTS

If A and B are dependent events, then the probability that both A and B occur is:

P(A and B) = __P(A) · P(B ½ A)__

Example 3

Find a conditional probability

Eye Color Find the probability that (a) a listed person has blue eyes and (b) a male has blue eyes.

	Green eyes	Blue eyes	Brown eyes	Hazel eyes
Male	27	35	15	23
Female	12	9	38	41

a. P(blue eyes) =

= _{_____} ^{= __0.22__}

b. P(blue eyes ½ male) =

= _{_____} = __0.35__

Example 4

Comparing independent and dependent events

You randomly select two marbles from a bag containing 15 yellow, 10 red, and 12 blue marbles. What is the probability that the first marble is yellow and the second marble is not yellow if (a) you replace the first marble before selecting the second, and (b) you do not replace the first marble?

Solution

Let A be “the first marble is yellow” and B be “the second marble is not yellow.”

a. If you replace the first marble before selecting the second marble, then A and B are _independent_ events. So, the probability is:

P(A and B) = __P(A) · P(B)__

= _{__________} = _{______} » _0.241_

b If you do not replace the first marble before selecting the second marble, then A and B are _dependent_ events. So, the probability is:

P(A and B) = __P(A) · P(B ½ A)_

= _{________} = _{______} » _0.248_

Example 5

Find probability of three dependent events

Pencils Your teacher passes around a box with 10 red pencils, 8 pink pencils, and 13 green pencils. If you and the two people in your group are the first to randomly select a pencil, what is the probability that all three of you select pink pencils?

Solution

Let A be that you choose a pink pencil, B be that the second group member chooses a pink pencil, and C be that the third group member chooses a pink pencil. These events are dependent. So, the probability is:

P(A and B and C) = __P(A) · P(B ½ A) · P(C ½ A and B)_

= _{____________} = _{________} » _0.0125_

Complete the following exercises.

3. Use the table in Example 3 to find the probability that a female has hazel eyes.

0.41

4. From Example 4, find the probability that both marbles will be yellow if (a) you replace the first marble and (b) you do not replace the first marble.

a. 0.164 b. 0.158

5. From Example 5, what is the probability that you and your group members all choose a red pencil?

0.0267