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Aptitude Question Discussion - PROBABILITY
Q.
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. What is the probability that it will rain on the day of Marie's wedding?
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.]
(Shouldn't the notation here be P(A1/B) and P(A2|B) the predictions have been made before it actually rains or not????)
Conditional probability P(A|B) = probability of A happening given that B)
P(B|A2)=0.1 [When it does not rain, the weatherman predicts rain 10% of the time.
To find the probability that it will rain on the day of Marie's wedding, we can use Bayes' theorem. Bayes' theorem allows us to calculate the probability of an event given some related information or conditions.
Let's use the notation P(A1) to represent the probability that it rains on Marie's wedding, P(A2) to represent the probability that it does not rain on Marie's wedding, P(B|A1) to represent the probability that the weatherman predicts rain given that it actually rains, and P(B|A2) to represent the probability that the weatherman predicts rain given that it does not rain.
The probability that it will rain on Marie's wedding can be calculated using Bayes' theorem as follows:
P(A1|B) = (P(B|A1) * P(A1)) / (P(B|A1) * P(A1) + P(B|A2) * P(A2))
Substituting the provided values, we have:
P(A1|B) = (0.9 * 5/365) / (0.9 * 5/365 + 0.1 * 360/365)
Simplifying the expression, we get:
P(A1|B) = 0.0405 / (0.0405 + 0.0986)
Calculating further, we get:
P(A1|B) ≈ 0.292
Therefore, the probability that it will rain on the day of Marie's wedding is approximately 0.292, or 29.2%.