Question

You are told that PA I B) = PBIA). Which statement below must be true?

Answer 1

Answer:

A and B are equally likely

Explanation:

The conditional probability is defined as follows:

$P(X|Y)=\frac{P(X\cap Y)}{P(Y)}$

where:

$P(X|Y)$ is the probability for X to occur, given that Y has occurred

$P(X\cap Y)$ is the probability that both X and Y occur at the same time

$P(Y)$ is the probability for Y to occur

Given two events A and B, we have:

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

and

$P(B|A)=\frac{P(A\cap B)}{P(A)}$

In this problem, we are told that

$P(A|B)=P(B|A)$

Which means that:

$\frac{P(A\cap B)}{P(B)}=\frac{P(A\cap B)}{P(A)}$

Simplifying, we get:

$P(A)=P(B)$

which means that A and B are equally likely.