Problem 3.

Let A and B be events with

$$P(A) \; = \; .5 , \quad P(B) \; = \; .2 , \quad P(A \cap B) \; = \; .1 .$$

(a) True or False: A and B are independent events. Justify your answer.

Solution: We check if the events satisfy the definition:

$$P(A) P(B) \; = \; .5 * .2 \; = \; .1 .$$

This equals $P(A \cap B)$, so ``Yes,'' the events are independent.

(b) Compute $P(A \cup B)$.

Solution: Using the formula in the Proposition on p. 62,

$$P(A \cup B) \; = \; P(A) + P(B) - P(A \cap B) \; = \; .5 + .2 - .1 \; = \; .6 .$$