Читать книгу Probability and Statistical Inference - Robert Bartoszynski - Страница 57

The best strategy for solving this kind of problem is finding the probabilities of all the intersections (in this case, ). The probability is given. Next, , and the events on the left are disjoint. So which means that , and hence, . Then, using , we have ; hence, . similarly . Applying formula (2.3), we have

For the case of three events: and , the same argument based on Venn diagrams gives the formula

(2.5)

It can be checked at Figure 2.7 that the formula (2.5) includes each part of the union exactly once.

Figure 2.7 Union of three events.

Formula (2.5) may be generalized to the case of the union of any finite number of events.

Theorem 2.4.1 For any events

(2.6)

Proof: We will proceed by induction. The theorem is true for . Assume now that formula (2.6) holds, and write

Since

we have

Applying formula (2.6) to and to , and then combining the corresponding terms, we obtain (2.6) with replaced by .