Addition Rule

Example 1

Suppose we roll a fair die. Find the probability of rolling a number larger than 3 or an even number.

Solution .

We have to find the number of outcomes favorable to our event and divide that by 6 which is the total number of outcomes that can occur when we roll a die.

Outcomes of 4, 5, and 6 are favorable to rolling a number larger than 3.

Outcomes of 2, 4, and 6 are favorable to rolling an even number.

There are four distinct outcomes favorable to rolling a number larger than 3 or an even number. These four outcomes are 2, 4, 5, and 6. Thus,

=

Note that if we had simply added P(x>3) to P(x is even) we would have gotten

= 1. This of course is incorrect because rolling a value more than 3 or an even number is a not a sure thing. We could get a one or a three when we roll the die. When we calculated we were actually counting two of the outcomes, getting a 4 and getting a 6, twice. The rule stated below takes this into account.

Addition Rule

If we use this rule on the problem in example one, we get

which gives us

=

Example 2

Assume that E and F are events with P(E) = 0.5, P(F) = 0.6, and P(E and F) = 0.23. Find P(E or F).

Solution .

Using the addition rule, we get

= 0.5 + 0.6 - 0.23 = 0.87.

Example 3

Roll a fair die. Find the probability of rolling a 3 or an even number.

Solution .

If x is the value rolled, then we want to find . This is found by using the addition rule. We get

= = .

In this last example the events "x=3" and "x is even" cannot occur simultaneously. These are examples of mutually exclusive events.

Definition .

Events A and B are called mutually exclusive if they both cannot occur simultaneously.

It should be noted that when A and B are mutually exclusive P(A and B) = 0.

Note that in example one, the events "x>3" and "x is even" are not mutually exclusive. They occur simultaneously when x=4 or x=6. We saw that which is not zero.

In example two events E and F are not mutually exclusive since P(E and F) was 0.23.

The events in example three are mutually exclusive since you can't roll a 3 and an even number at the same time. Also note that

Example 4

The following table shows the number of managers, servers, and cooks at a fast food restaurant broken down by age.

a.) If an employee is selected at random, find the probability that the individual is a server (S) or a cook (C).

Solution .

Note that there are 55 employees with 40 servers and 10 cooks.

= =

b.) If an employee is selected at random, find the probability that the person is a manager (M) or between the ages of 50 and 59 ( ).

Solution .

Note that there are five managers, six people between 50 and 59, and two people who are managers and are between 50 and 59.

= =