# 6.2. Mutually Exclusive Events and the Addition Rule

**Mutually Exclusive Events and the Addition Rule**

In the previous chapter, we learned to find the union, intersection, and complement of a set. We will now use these set operations to describe events.

The **union** of two events *E* and *F*, , is the set of outcomes that are in *E* or in *F* or in both.

The **intersection** of two events *E* and *F*, , is the set of outcomes that are in both *E* and *F*.

The **complement** of an event *E*, denoted by E^{c}, is the set of outcomes in the sample space *S* that are not in *E*. It is worth noting that *P*(*E*^{c}) = 1 − *P* (*E*). This follows from the fact that if the sample space has *n* elements and *E* has *k* elements, then E^{c} has *n *− *k* elements. Therefore:

Of particular interest to us are the events whose outcomes do not overlap. We call these events mutually exclusive.

Two events *E* and *F* are said to be **mutually exclusive** if they do not intersect. That is, =∅.

Next we’ll determine whether a given pair of events are mutually exclusive.

Example 6.2.1

*E*= {The card drawn is an Ace}

*F*= {The card drawn is a heart}

**Solution**

*E*and

*F*are not mutually exclusive.

Example 6.2.2

*G*= {The sum of the faces is six}

*H*= {One die shows a four}

**Solution**

*G*= {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

*H*= {(2, 4), (4, 2)}

Example 6.2.3

**Solution**

*M*={BBB , BBG , BGB , BGG , GBB , GBG , GGB} and

*N*={GGG}

*M*and

*N*are mutually exclusive.

We will now consider problems that involve the union of two events.

Example 6.2.4

**Solution**

*E*be the event that the number shown on the die is an even number, and let

*F*be the event that the number shown is greater than four.

*S*={1, 2, 3, 4, 5, 6}. The event

*E*={2, 4, 6}, and the event

*F*={5, 6}

*E*and

*F*has been counted twice, once as an element of

*E*and once as an element of

*F*. In other words, the set has only four elements and not five. Therefore, and not .

*S*, the events

*E*and

*F*, and are listed below.

*S* = {1, 2, 3, 4, 5, 6} , *E* = {2, 4, 6} , *F* = {5, 6} , and = {6}.

*S*,

*E*,

*F*, and .

*E*will happen, or

*F*will happen, or both will happen. If we count the number of elements

*n*(

*E*) in

*E*, and add to it the number of elements

*n*(

*F*) in

*F*, the points in both

*E*and

*F*are counted twice, once as elements of

*E*and once as elements of

*F*. Now if we subtract from the sum,

*n*(

*E*) +

*n*(

*F*), the number , we remove the duplicity and get the correct answer. So as a rule:

*P*(

*E*) and

*P*(

*F*), we have added twice. Therefore, we must subtract , once.

**Addition Rule**, for finding the probability of the union of two events. It states:

*E*and

*F*are mutually exclusive, then and , and we get:

Example 6.2.5

**Solution**

*A*be the event that the card is an ace, and

*H*the event that it is a heart. Since there are four aces, and 13 hearts in the deck,

*P*(

*A*) = 4/52 and

*P*(

*H*) = 13/52.

Example 6.2.6

*F*and

*T*are as follows:

*F*= {The sum of the dice is four} and

*T*= {At least one die shows a three}

**Solution**

*F*and

*T*, and as follows:

*F*= {(1, 3), (2, 2), (3, 1)}

*T*= {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)}

Example 6.2.7

**Solution**

*A*be the event that the board approves the position, and

*S*be the event that Mr. Washington gets selected. We have:

Example 6.2.8

**Solution**

*C*be the event that the weekend will be cold, and

*R*be event that it will be rainy. We are given that:

We summarize this section by listing the important rules.

**The Addition Rule**: For two events*E*and*F*,**The Addition Rule for Mutually Exclusive Events**: If two events*E*and*F*are mutually exclusive, then**The Complement Rule**: If*E*is the complement of event^{c}*E*, then

**Practice questions**

**1.** Determine whether the following pairs of events are mutually exclusive:

**a.** Three coins are tossed. *A = *{Two heads come up}, *B* = {At least one tail comes up}.

**b. **Two dice a rolled. *C = *{The sum of the dice is 9}, *D* = {At least one dice shows a 2}.

**c. ***E = *{You will get an A on your next exam}, *F* = {You will pass your next exam}.

**2. **Two dice are rolled, and the events *G* and *H* are as follows. *G = *{The sum of the dice is 8}, *H* = {Exactly one die shows a 6}. Use the addition rule to find .

**3. **At Ryerson University, 20% of the students take a Mathematics course, 30% take a Statistics course, and 10% take both. What percentage of students take either a Mathematics or Statistics course?

**4. **The following table shows the distribution of coffee drinkers by gender:

Coffee drinker | Males (M) | Females (F) | TOTAL |

Yes (Y) | 31 | 33 | 64 |

No (N) | 19 | 17 | 36 |

50 | 50 | 100 |

Use the table to determine the following probabilities:

**a**.

**b.**

**5. **If , , and , use the addition rule to find .

**6. **A provincial park has 240 campsites. A total of 90 sites have electricity. Of the 66 sites on the lakeshore, 24 of them have electricity. If a site is selected at random, what is the probability that:

**a**. It will have electricity?

**b.** It will have electricity or be on the lakeshore?

**c.** It will be on the lakeshore and not have electricity?