Question: 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

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. If the forecaster announces a probability p that the summer will be rainy, she 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 her expected profit by announcing that the probability of a rainy summer is 0.6.

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