## Example 327

A stock market analyst examined the prospects of the shares of a large number of corporations. When the performance of these stocks was investigated one year later, it turned out that 25% performed much better than the market average, 25% much worse, and the remaining 50% about the same as the average. Forty percent of the stocks that turned out to do much better than the market were rated "good buys" by the analyst, as were 20% of those that did about as well as the market and 10% of those that did much worse. What is the probability that a stock rated a "good buy" by the analyst performed much better than the market average?

Define the following events:

E,: Stock performs much better than the market average.

E2: Stock performs about the same as market average.

Ey\ Stock performs much worse than market average.

A: Stock is rated "good buy" by the analyst.

From the statement of the example, we have the probabilities

P(E,) = .25 P(E2) = . 50 P(Ei) = . 25 and the conditional probabilities

It is required to find the probability that a stock performs much better than the market average, given that it was rated a "good buy" by the analyst. This is the conditional probability P(£i | A), which is obtained from Bayes' theorem as follows:

P(A |£|)£(£i) + P(A\E2)P(E2) + P(A\E,)P(E,) (.4)(.25)