Each day, a weather forecaster predicts whether or not it will rain. For 80% of rainy days, she correctly predicts that it will rain. For 90% of non-rainy days, she correctly predicts that it will not rain. Suppose that 10% of days are rainy and 90% are non-rainy.

Question

a. What proportion of the forecasts are correct?
b. Another forecaster always predicts that there will be no rain. What proportion of these forecasts are correct?

Answer 1

Let R denote the event of a rainy day, and let C denote the event that the forecast is correct. Then P(R) = 0.1, P(C|R) = 0.8, and = 0.9.



(a) P(C)=P(C|R)P(R) +

=(0.8)(0.1) + (0.9)(1 ? 0.1)

=0.89



(b) A forecast of no rain will be correct on every non-rainy day. Therefore the probability is 0.9.