aProblem 27E

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.

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?

Step 1 of 3:

The experiment given here is of weather forecasting.

Let us define event as a rainy day and event as non rainy day.

And let event C denotes that the prediction i9s correct.

It is given that 10% of the days are rainy days. This implies P()=0.1.Also given that 90% of the days are non rainy days. This implies P()=0.9.

And also it is given that P(correct prediction|it is a rainy day)=0.8 and P(correct prediction|it is a non rainy day)=0.9

That is P(C|)=0.8 and P(C|)=0.9.

Using these probability values we have to find the remaining probabilities.

Step 2 of 3:

(a)

Here we have to find the proportion of correct forecast. It is denoted by P(C).

This probability is given by,

P(C)=P(C|)*P()+ P(C|)*P()

=(0.8*0.1)+(0.9*0.9)

=0.08+0.81

=0.89

Thus, the proportion of correct forecast is