Bunuel wrote:
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?
A. 1/9
B. 2/9
C. 1/3
D. 3/4
E. 5/9
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions Solution:This problem can be solved by the application of Bayes’ theorem, but a more intuitive solution is provided here.
Let’s first look at the 4 different possible outcomes that can happen concerning whether it rains (R) on Maire’s wedding day or it doesn’t rain (nR), and whether the weather status was predicted (P) or was not predicted (nP). We have:
(R, P) or (R, nP) or (nR, P) or (nR, nP)
Let’s calculate the probability of each outcome:
P(R, P) = 5/365 x 0.9 = 4.5/365
P(R, nP) = 5/365 x 0.1 = 0.5/365
P(nR, P) = 360/365 x 0.1 = 36/365
P(nR, nP) = 360/365 x 0.9 = 324/365
We see that only 2 of the 4 possible outcomes deal with the event “the weatherman has predicted rain for tomorrow,” and they are the following outcomes: (R,P) and (nR,P), and the total probability that either of these 2 outcomes will happen is the sum of their individual probabilities, which is 4.5/365 + 36/365 = 40.5/365.
But, out of these two outcomes, only (R, P), with probability 4.5/365, is relevant to answering the question “what is the probability that it will rain on Marie’s wedding day?” Thus, the probability that it will rain on Marie’s wedding day, given that the weatherman has predicted rain, is 4.5/365 / 40.5/365 = 4.5/40.5 = 45 / 405 = 1/9.
Answer: A