Your going to garden May 1st.  In recent years, it has rained on average of 30% of the days you will be gardening.  On days when it rains, 85% of the time the rain was forecasted, on days when it doesnt rain only 20% of the time was rain forecasted.  What is the probability that it rains on the day your going to garden given that the weather forecast predicts rain on that day?

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Your going to garden May 1st.  In recent years, it has rained on average of 30% of the days you will be gardening.  On days when it rains, 85% of the time the rain was forecasted, on days when it doesnt rain only 20% of the time was rain forecasted.  What is the probability that it rains on the day your going to garden given that the weather forecast predicts rain on that day?

Expert Solution
Step 1

Given information:

PRain=0.30P(No rain)=1-P(Rain)=1-0.30=0.70

PForecastedRain=0.85

PForecastedNo rain=0.20

Step 2

Use the Bayes' theorem to determine the required probability:

PRainForecasted=PForecastedRain×PRainPForecastedRain×PRain+PForecastedNo rain×PNo rain=0.85×0.300.85×0.30+0.20×0.70=0.2550.255+0.14=0.2550.395=0.6456

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