In a city, Angelica has a 60% of chance to be late for her work if it rains. If it doesn't, she has only a 30% of chance to be late for her work.
One day Angelica saw on the TV that there was 25% of chance of rain on that day.
Knowing that Angelica did not arrive late at her work on that day, what is the probability that it rained that day?
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By the Bayes' Theorem we have:
P(rained|not late) = P(not late) P(rained and not late)
P(rained|not late) = P(rained) x P(not late|rained) + P(not rained) x P(not late|not rained)) P(rained) x P(not late|rained)
P(rained|not late) = 2 5 % × 4 0 % + 7 5 % × 7 0 % 2 5 % × 4 0 %
P(rained|not late) = 6 2 5 0 % % 1 0 0 0 % %
P(rained|not late) = 0 . 1 6 = 1 6 %