\(P(A | B) = P(A \cap B) / P(B)\) and \(P(B | A) = P(B \cap A) / P(A)\)

we can rewrite the equality in the question to be:

  1. \(P(A \cap B)P(A) = P(B \cap A)P(B)\) using cross-multiplication.

thus we can divide the \(P(A \cap B)\) on both sides to get

  1. \(P(A) = P(B)\)

of \(P(A \cup B)\) that is the probability of either of the two events A and B happening