Compute the following conditional probabilities

Question

independent events E and F in a sample space S, but assume that Pr[E] = 0.4 and Pr[F] = 0.15.
Compute the following conditional probabilities:

(1) Pr[E′| F] =

(2) Pr[E | F′] =

Answer 1

Events A and B are independent if the probability of A happening is not affected by whether or not event B has happened. In other words, A and B are independent if Pr[ A | B ] = Pr[ A ]. (This is equivalent to saying Pr[ A ∩ B ] = Pr[ A ] · Pr[ B ].)

Furthermore, If A and B are independent, then so are A and B', A' and B, and A' and B'.

Therefore, for Part (2), Pr[ E | F' ] is equal to Pr[ E ] which is 0.4.

For Part (1), we use the fact that for any event A, probability that A does not happen (i.e., the event A' does happen) is Pr[ A' ] = 1 - Pr[ A ].

This means Pr[ E' ] = 1 - Pr[ E ] = 1 - 0.4 = 0.6, and thus

Pr[ E' | F ] = Pr[ E' ] = 0.6.