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

A and B are independent if:

P r (A \cap B) = P r (A) + P r (B)

If $P r (B) > 0$ ,

P r (A ∣ B) = \frac{P r ( A \cap B )}{P r ( B )} = P r (A)

Bayes Theorem

P r (B ∣ A) = \frac{P r ( B ) P r ( A ∣ B )}{P r ( B ) P r ( A ∣ B ) + P r ( B ^{C} ) P r ( A ∣ B ^{C} )}

Luca's Notes