The forecast says rain, should you believe it?

Question

In a dry region, it actually rains on only 20% of days. A local forecaster is fairly good: when it truly will rain, they call for rain 90% of the time; but on dry days they still cry "rain" 30% of the time (false alarms).

Today the forecaster calls for rain.

What is the probability it actually rains today?

Accepted Answer

Let R mean "it rains" and "call" mean the forecaster predicts rain. Use 100 typical days. - Rainy days: 20; the forecaster calls rain on 90\% \Rightarrow 18. - Dry days: 80; the forecaster calls rain on 30\% \Rightarrow 24. The forecaster calls "rain" on 18 + 24 = 42 days, and it actually rains on 18 of them: P(R \mid 	ext{call}) = \frac{18}{42} = \frac{3}{7} \approx 0.429. Even after a rain forecast, it's more likely than not to stay dry, because dry days are so common that their false alarms outnumber the true calls. Takeaway. A forecast's reliability depends heavily on the base rate. In a…

The forecast says rain, should you believe it?

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