Chapter 11.4, Problem 20PPSE

Solution

(a)

To calculate: The probability that both events A and B occur at the same time $P(A\cap B)$ .

  $P(A\cap B)=0.15$

Given:

A and B are two independent events.

  $\begin{array}{l}P(A)=0.60\\ P(B)=0.25\end{array}$

Formula used:

When two events A and B are independent then their probability of intersection is calculated using the formula

  $P(A\cap B)=P(A)\times P(B)$

Calculation:

The probability is calculated as shown below

  $\begin{array}{c}P(A\cap B)=P(A)\times P(B)\\ =0.60\times 0.25\\ =0.15\end{array}$

The probability for the intersection of two events A and B is $P(A\cap B)=0.15$

(b)

To calculate:

The conditional probability that event A occurs given that B has already occurred $P(A|B)$ .

  $P(A|B)=0.60$

Formula used:

The conditional probability that event a occur where event B has already occurred is calculated using the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$ .

Calculation:

From previous part, the probability of intersection of A and B is $P(A\cap B)=0.15$ . The conditional probability is calculated as shown below:

  $\begin{array}{c}P(A|B)=\frac{P(A\cap B)}{P(B)}\\ =\frac{0.15}{0.25}\\ =0.60\end{array}$

The conditional probability is $P(A|B)=0.60$ .

(c)

To compare: The conditional probability that event A occurs given that B has already occurred and the probability that event A occur.

Given:

The conditional probability that event a occur where event B has already occurred is $P(A|B)=0.60$

The probability that event A occurs is $P(A)=0.60$

Interpretation:

it is observed that $P(A|B)=P(A)=0.60$ , the probability that event a occur given that event B has already occurred is the same as the probability that event A occurs.

(d)

To explain: The independent events where occurrence of event B has no effect on probability of A.

Interpretation:

It is observed that $P(A|B)=P(A)=0.60$ , the probability that event a occur given that event B has already occurred is the same as the probability that event A occurs.

Thus, it can be said that the occurrence of event B has no impact on the occurrence of event A.

So, they are considered to be independent. As the conditional probability of A given that B already occurred is same as probability of A, it can be concluded that A and B are independent events.