Given: \(P(A|B) = P(A'|B)\)

Using the definition of conditional probability, we have:

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

Since \(P(B)\) is in the denominator on both sides, we can multiply both sides by \(P(B)\), assuming \(P(B) \neq 0\):

\(P(A \cap B) = P(A' \cap B)\)

We know that \(A' = \Omega - A\), where \(\Omega\) is the sample space. Therefore, \(A' \cap B = B - (A \cap B)\). So, we can write \(P(A' \cap B) = P(B) - P(A \cap B)\)

Substituting this back into the equation, we get:

\(P(A \cap B) = P(B) - P(A \cap B)\)

Adding \(P(A \cap B)\) to both sides:

\(2 P(A \cap B) = P(B)\)

Dividing both sides by 2:

\(P(A \cap B) = \frac{1}{2} P(B)\)