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:
Here we have to find the proportion of correct forecast. It is denoted by P(C).
This probability is given by,
Thus, the proportion of correct forecast is