Chapter 13.5, Problem 43E

To determine

To prove:The events A and B are independent events.

Explanation of Solution

Given information:

The probability of an event A is equal to probability of same event, provided that event B has already occurred.

Formula used:

Given an event B , the conditional probability of event A is $P\left(A|B\right)=\frac{P\left(A\text{ and B}\right)}{P\left(B\right)}$ where $P\left(B\right)\ne 0$ .

Proof:

Consider the provided information that probability of an event A is equal to probability of same event, provided that event B has already occurred.

Probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Consider two events A and Bsuch that they are independent events. Then $P\left(A\text{ and B}\right)=P\left(A\right)\cdot P\left(B\right)$ .

Given an event B , the conditional probability of event A is $P\left(A|B\right)=\frac{P\left(A\text{ and B}\right)}{P\left(B\right)}$ where $P\left(B\right)\ne 0$ .

According to question $P\left(A|B\right)=P\left(A\right)$ .

Therefore,

  $\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and B}\right)}{P\left(B\right)}\\ P\left(A\right)=\frac{P\left(A\text{ and B}\right)}{P\left(B\right)}\\ P\left(A\text{ and B}\right)=P\left(A\right)\cdot P\left(B\right)\end{array}$

That is events A and B are independent events.

Hence, it is proved that if probability of an event A is equal to probability of same event, provided that event B has already occurred then A and B are independent events