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Understanding Uncertainty, Revised Edition by Dennis V. Lindley

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Chapter 8

Three Events

8.1 The Rules of Probability

So far in this book we have almost entirely been concerned with studies involving only two events. The ideas developed there are now extended to situations with three or more events. The rules of probability that were developed in Chapter 5 are perfectly adequate to deal with the extension, and no new rules are required, but they do lead to some surprising results when more events are contemplated. We begin by looking again at the three rules.

The convexity rule in §5.4 deals with a single event and requires no elaboration. The addition rule, in the simpler form of Equation (5.2) of §5.2, says that if two events, E and F, are exclusive (that is, cannot both be true) then

(8.1) equation

The extension to three is immediate. Suppose E, F, and G are three events that are exclusive, in the sense that no two of them can both be true, then

(8.2) equation

where E or F or G means the event that is true if, and only if, one of them is true. There are several ways to see that this is correct. One is to return to the urn and suppose that some balls are emerald, E, some fawn, F, some green, G, and the remainder without color. A colored ball has only one color, so the colors are exclusive and the total number of colored balls is the sum of the numbers that are ...

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