# The events A and B are mutually exclusive. Suppose P(A)=0.23 and P(B)=0.26 a. What is the...

## Question:

The events {eq}A {/eq} and {eq}B {/eq} are mutually exclusive. Suppose {eq}P(A)=0.23 {/eq} and {eq}P(B)=0.26. {/eq}

a. What is the probability of either {eq}A {/eq} or {eq}B {/eq} occurring?

b. What is the probability that neither {eq}A {/eq} nor {eq}B {/eq} will happen?

## Mutually Exclusive Events:

Mutually exclusive events are a set of events that cannot occur at the same time. For example, the event when a coin toss results in heads and the event when the same toss results in tails are mutually exclusive. If {eq}A {/eq} and {eq}B {/eq} are mutually exclusive events, then the probability of their intersection is zero, that is, {eq}P(A \cap B) = 0 {/eq}. Because of this, we can calculate the probability of either mutually exclusive events happening just by adding each probability.

## Answer and Explanation:

(a) If either {eq}A {/eq} or {eq}B {/eq} occur, then we have to consider the union of the two events. That is, we want to calculate {eq}P(A \cup B) {/eq}. Using the Inclusion-Exclusion Principle, we can rewrite this expression as $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Note that we are given the first two terms. For the third term, recall that {eq}A {/eq} and {eq}B {/eq} are mutually exclusive. This means that they cannot occur at the same time, or {eq}P(A \cap B) = 0 {/eq}. We can then evaluate the desired probability as \begin{align*} P(A \cup B) &= P(A) + P(B) - P(A \cap B) \\ &= 0.23 + 0.26 - 0 \\ &= 0.49 \end{align*}

(b) If neither {eq}A {/eq} nor {eq}B {/eq} occur, then we have to consider the event when either of them occurs and take the complement of that event. From (a), that event is {eq}A \cup B {/eq}. Thus, we want to calculate {eq}P((A \cup B)^\complement) {/eq}. From the properties of complementary events, we can calculate this probability as \begin{align*} P((A \cup B)^\complement) &= 1 - P(A \cup B) \\ &= 1 - 0.49 & \text{from (a)} \\ &= 0.51 \end{align*}