Marie is getting married tomorrow, at an outdoor ceremony in the desert. In rece

Question

Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time.
a)What is the probability that it will rain on the day of Marie's wedding?
b)If the weatherman forecast rain, find the probability that it will rain on the day of Marie's wedding.

Explanation / Answer

proability of rain on a given day i.e P(rain) = 5/365 = 0.0137

P(no rain) = 1 - 0.0137 = 0.9863

P(weatherman forecasts rain / rains) = 0.9 => P(weatherman forecasts rain and rains)/P(rains) = 0.9

P(weatherman forecasts rain and rains) = 0.9*P(rains) = 0.9*0.0137 = 0.01233

P(weatherman forecasts rain/doesn't rain) = 0.1 =>

P(weatherman forecasts rain and doesn't rain)/P(no rain) = 0.1

P(weatherman forecasts rain and doesn't rain) = 0.1*0.9863 = 0.0986

P(weatherman forecasts rain) = P(weatherman forecasts rain and doesn't rain) + P(weatherman forecasts rain and doesn't rain)

= 0.01233+ 0.0986 = 0.11093

a)

probability that it will rain on the day of Marie's wedding = proability of rain on a given day = 0.0137

b)

probability that it will rain on the day of Marie's wedding given weatherman forecast rain

= P(rain / weatherman forecasts rain)

By Bayes theorem we know that

P(rain / weatherman forecasts rain) = P(rain and weatherman forecasts rain)/P(weatherman forecasts rain)

= 0.01233/0.11093 = 0.1112

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