Suppose that you are hiring a weather forecaster to predict the probability that next summer will be rainy or sunny. The following suggests a method that can be used to ensure that the forecaster is...

Suppose that you are hiring a weather forecaster to predict the probability that next summer will be rainy or sunny. The following suggests a method that can be used to ensure that the forecaster is accurate. Suppose that the actual probability of next summer being rainy is 0.6. (For simplicity, we assume that the summer can only be rainy or sunny.) If the forecaster announces a probability p that the summer will be rainy, he receives a payment of 1 - (1 - p)²
if the summer is rainy and a payment of 1 - p²
if the summer is sunny. Show that the forecaster will maximize his expected profit by announcing that the probability of a rainy summer is 0.6.