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  Let A and B be events in a sample space, and suppose

$$P(A) \; = \; .4, \quad P(B) \; = \; .5 .$$

Calculate $P(A \cup B)$ if

(a) A and B are independent.

Solution: The general formula is

$$P(A \cup B) \; = \; P(A) + P(B) - P(A \cap B) .$$

When A and B are independent

$$P(A \cap B) \; = \; P(A) P(B) \; = \; .5 * .4 \; = \; .2 .$$

Hence

$$P(A \cup B) \; = \; .4 + .5 - .2 \; = \; .7 .$$

(b) A and B are mutually exclusive.

Solution: Axiom 3 of probability says that when A and B are mutually exclusive

$$P(A \cup B) \; = \; P(A) + P(B) .$$

Hence

$$P(A \cup B) \; = \; .4 + .5 \; = \; .9 .$$

Problems? Questions? (I may have made a mistake, so don't hestitate to ask) dcox@stat.rice.edu