10.4 Find Probabilities of Disjoint and Overlapping Events

VOCABULARY

Compound Event

The union or intersection of two events

Disjoint or mutually exclusive events

Two events that have no outcomes in common

PROBABILITY OF COMPOUND EVENTS

If A and B are two events, then the probability of A or B is:
P(A or B) = _P(A) + P(B) - P(A and B)_

If A and B are disjoint events, then the probability of A or B is:
P(A or B) = _P(A) + P(B)_

Example 1

Find probability of disjoint events

You roll a six-sided number cube. What is the probability of rolling a 2 or a 5?

Let event A be rolling a 2 and event B be rolling a 5. A has _1_ outcome and B has _1_ outcome. Because A and B are mutually exclusive, the probability is:

P(A or B) = _P(A) + P(B)_ = +

= = » _0.333_

Complete the following exercise.

1. You roll a six-sided number cube. What is the probability of rolling a 1 or an even number?

» 0.667

Example 2

Find probability of compound events

You roll a six-sided number cube. What is the probability of rolling an odd number or a number less than 3?

Solution

Let event A be rolling an odd number and event B be rolling a number less than 3. A has __3__ outcomes and B has __2__ outcomes. Of these, __1__ outcome is common to A and B. So, the probability of rolling an odd number or a number less than 3 is:

P(A or B) = _P(A) + P(B) - P(A and B)_

=

Example 3

Use a formula to find P(A and B)

Music In a survey of 300 students, 150 like pop music or country music. There are 97 students who like pop music and there are 83 students who like country music. What is the probability that a randomly selected student likes both pop music and country music.

Solution

Let event A be selecting a student who likes pop music and event B be selecting a student who likes country music. From the given information you know that:

P(A)= , P(B)= ,and P(A or B) =

The probability that a randomly selected student likes both pop music and country music is P(A and B).

P(A or B)=__P(A) + P(B) - P(A and B)__ Write general formula.

= Substitute

P(A and B) = Solve for P(A and B)

P(A and B) = = _0.10_ Simplify.

Example 4

Find the probability of complements

When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that the sum is not 4 and the sum is greater than or equal to 3.

Solution

a. P(sum is not 4) = __1 -P(sum is 4)__

= »__0.917__

b. P(sum ³ 3) = __1-P(sum < 3)__

= »__0.972__

PROBABILITY OF THE COMPLEMENT OF AN EVENT

The probability of the complement of A is P( ) = _1 - P(A)_.

Complete the following exercises.

2. You roll a six-sided number cube. What is the probability of rolling a number less than 4 or an even number?

3. In a survey of 125 people, 90 of them like orange juice or grape juice. There are 62 people who like orange juice and 43 people who like grape juice. What is the probability that a randomly selected person likes both orange juice and grape juice?

4. From Example 4, find the probability that the sum is not 8.