Alice is getting married on May 5th at an outdoor ceremony in a desert. In recent years it has rained only 5 days each year. Unfortunately the weather person has predicted rain for tomorrow. When it actually rains the weather person correctly forecasts rain 90% of the time. When it doesn’t rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Alice’ wedding?
Answers
Answer:
0.111
Step-by-step explanation:
By Bayes Theorem
Event A1 : It rains on Marie's wedding.
Event A2 : It does not rain on Marie's wedding.
Event B : The weatherman predicts rain.
• In terms of probabilities, we know the following:
- P(A1 ) = 5/365 =0.0136985 [It rains 5 days out of the year.]
> P(A2 ) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.)
- P( B | A1) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]
- P(B |A2 ) = 0.1 [When it does not rain, the weatherman
predicts rain 10% of the time.)
We want to know P( A, | B ), the probability it will rain on the day of Marie's wedding, given a forecast for rain by the weatherman. The answer can be determined from Bayes' theorem, as shown below.
P(A1 | B) = [P(A1) * P(B | A1)] / [P(A1)*P(B | A1) + P(A2)*P(B | A2)]
P(A1 | B) = (0.014)(0.9)/(0.014)(0.9) + (0.986) (0.1)
P(ATB) = 0.111
Given : Alice is getting married on May 5th at an outdoor ceremony in a desert. In recent years it has rained only 5 days each year.
weather person has predicted rain for Wedding day
When it actually rains the weather person correctly forecasts rain 90% of the time.
When it doesn’t rain, he incorrectly forecasts rain 10%
To Find : probability that it will rain on the day of Alice’ wedding?
Solution:
Rain 5 days in a year
Probability of Rain on any Day = 5/365 = 1/73
Rain Happens on wedding Day = (1/73)
weather person has predicted rain for tomorrow. When it actually rains the
weather person correctly forecasts rain 90% of the time.
= (1/73) (0.9)
= 0.9/73
Rain Does not Happens on wedding Day = 1 - (1/73) = 72/73
When it doesn’t rain, he incorrectly forecasts rain 10%?
= (72/73) (0.1)
= 7.2/73
Probability that Rain Happens on wedding Day = 0.9 / (0.9 + 7.2)
= 0.9/8.1
= 1/9
= 0.111
Probability that Rain Happens on wedding Day = 1/9 = 0.111
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