Question

On the probability field {Ω, K, P}, consider two independent events A and B with P(A) = 1/2, P(B) = 2/5. Calculate the conditional probability that both A and B occur, given that at least one of the events A and B has occurred.

Answer 1

$P(\frac{both \; A \;and\; B \;occur}{at \; least \; A or \; B \;occured}) = P(\frac{A \cap B}{A \cup B}) \\ = \frac{P[(A \cap B) \cap (A \cup B)]}{P(A \cup B)} \\ = \frac{P(A \cap B)}{P(A \cup B)} \\ = \frac{P(A) \times P(B)}{P(A) + P(B) -P(A \cap B)} \\ = \frac{\frac{1}{2} \times \frac{2}{5}}{\frac{1}{2} + \frac{2}{5} - \frac{2}{10}} \\ = \frac{2}{7}$