Find the Probability

Probability Level 2

Let $A$ and $B$ be events such that $P(A) = \frac12 = P(B)$ and $P(A^c \cap B^c) = \frac13$ . Find the probability of the event $A^c \cup B^c$ .

1 solution

Tom Engelsman
Apr 10, 2020

Knowing that $P(A^{c} \cup B^{c}) = P(A^{c}) + P(B^{c}) - P(A^{c} \cap B^{c})$ , the required probability computes to:

$P(A^{c} \cup B^{c}) = (1 - \frac{1}{2}) + (1 - \frac{1}{2}) - \frac{1}{3} = \boxed{\frac{2}{3}}.$

The events don't have to be independent. The identity holds for any pair of events. It's the Inclusion-Exclusion Principle.

Richard Desper - 1 year, 2 months ago

Edited & done, Richard, thanks.

tom engelsman - 1 year, 2 months ago