Q. 2.8 - A First Course in Probability

Question

Suppose that A and B are mutually exclusive events for which $P\left(A\right) = .3 \text{and} P\left(B\right) = .5$. What is the probability that

(a) either A or B occurs?

(b) A occurs but B does not?

(c) both A and B occur?

Step-by-Step Solution

Answer

(a) The probability that either A or B occurs is $0.8$.

(b) The probability that A occurs but B does not is $0.3$.

(c) The probability that both A and B occur is $0$.

Part (a) Step 1. Given information.

It is given that,

$$\begin{gathered} P\left(A\right) = 0.3 \\ P\left(B\right) = 0.5 \end{gathered}$$

Let us consider that A and B are two events that are mutually exclusive, so $A\cap B=0$.

Part (a) Step 2. Find the probability that either A or B occurs.

Probability that either A or B occurs $P(A\cup B)$

$$\begin{gathered} P(A\cup B) = P\left(A\right) + P\left(B\right) - P(A\cap B) \\ =0.3 +\hspace{0.17em}0.5 - 0 \\ =0.8 \end{gathered}$$

Therefore, $P(A\cup B)=0.8$.

Part (b) Step 1. Find the probability that A occurs but B does not.

Probability that A occurs but B does not $=P(A\cap \overline{B})$

$$\begin{gathered} P(A\cap \overline{B}) = P\left(A\right) - P(A\cap B) \\ =0.3 - 0 \\ =0.3 \end{gathered}$$

Therefore, $P(A\cap \overline{B})=0.3$.

Part (c) Step 1. Find the probability that both A and B occur.

Probability that both A and B occur $=P\left(A\cap B\right)$

As A and B are two events that are mutually exclusive, then the probability of intersection of these events is $0$.

Therefore, $P\left(A\cap B\right)=0$