Today, I see high flow conditions in the river, which means it rained somewhere

Question

Today, I see high flow conditions in the river, which means it rained somewhere in the headwater basins. What is the probability that it rained in the headwater basins of the Elm Fork? For simplicity, ignore the Clear Fork and assume that there are only two groups of headwater basins, one that drains into the Elm Fork and the other that drains into the West Fork. High flow conditions can occur only from rainfall in the headwater basins. Assume that occurrence of rainfall in the headwater basins of the Elm Fork is independent of that of the West Fork. Use the following probabilities which are estimated from a long record of historical observations:

Prob[it rains in the headwater basins of the Elm Fork]=0.05

Prob[it rains in the headwater basins of the West Fork]=0.04

Prob[high flow occurs at Dallas | it rained in the headwater basins of the Elm Fork]=0.25

Prob[high flow occurs at Dallas | it did not rain in the headwater basins of the Elm Fork, it rained in the headwater basins of the West Fork]=0.2

Explanation / Answer

given information is

Prob[it rains in the headwater basins of the Elm Fork]=0.05

Prob[it rains in the headwater basins of the West Fork]=0.04

Prob[high flow occurs at Dallas | it rained in the headwater basins of the Elm Fork]=0.25

Prob[high flow occurs at Dallas | it did not rain in the headwater basins of the Elm Fork, it rained in the headwater basins of the West Fork]=0.2

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now question is

What is the probability that it rained in the headwater basins of the Elm Fork?

Prob[it rains in the headwater basins of the Elm Fork]=0.05

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