Chapter 11, Problem 28E

Solution

To calculate: The conditional probability that events A, and B occurs together given that event A or B has already occurred $P(A\cap B|A\cup B)$

The conditional probability that events A, and B occurs together given that event A or B has already occurred $P(A\cap B|A\cup B)=0.125$ .

Given:A and B are two events where

  $\begin{array}{l}P(A)=0.5\\ P(B)=0.4\\ P(A\cap B)=0.1\end{array}$

Concept used:

The conditional probability of any two events A, B is calculated using the formula shown below

  $P(A|B)=\frac{P(A\cap B)}{P(B)}$ .

The probability of the union of two events A and B is calculated using the formula

  $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Calculation:

The probability that both events A or B occur is calculated as shown below

  $\begin{array}{l}P(A\cup B)=P(A)+P(B)-P(A\cap B)\\ P(A\cup B)=0.5+0.4-0.1\\ P(A\cup B)=0.8\end{array}$

The intersection of events $(A\cap B)$ and $(A\cup B)$ is always $(A\cap B)$ . As $(A\cap B)$ is a subset of $(A\cup B)$ . Thus it can be said that

  $P\left((A\cap B)\cap (A\cup B)\right)=P(A\cap B)$

The conditional probability is calculated as shown below

  $\begin{array}{c}P(A\cap B|A\cup B)=\frac{P\left((A\cap B)\cap (A\cup B)\right)}{P(A\cup B)}\\ P(A\cap B|A\cup B)=\frac{P(A\cap B)}{P(A\cup B)}\\ P(A\cap B|A\cup B)=\frac{0.1}{0.8}=0.125\end{array}$ .

The conditional probability that events A, and B occurs together given that event A or B has already occurred $P(A\cap B|A\cup B)=0.125$ .