23. Each night different meteorologists give us the “probability” that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of

We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather.
Suppose now that a given meteorologist is aware of this and so wants to maximize his or her expected score. If this individual truly believes that it will rain tomorrow with probability p∗, what value of p should he or she assert so as to maximize the expected score?

1-(1-p) 1 (1p) 1-p if it does rain if it does not rain