Bayes Theorem
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Given a conditional probability of P(A | B), we could apply the Bayes Theorem with P(B | A). It's essentially the reverse of a conditional probability.
Scenario: Given the following probability tree, find the probability that a student did not wear a hat, given that they received a bid from Delta Apple Pi.
Our probability tree goes from "did / did not wear hat" ➡️ "received / did not receive bid"... not the other way around!
Event A = received a bid
Event B = did not wear a hat
P(B | A) = [P(A | B) x P(B)] / P(A)
P(B) = 0.60
P(A | B) = 0.20
P(A) = 0.30 + 0.12
P(A) = 0.42
P(B | A) = [0.20 x 0.60] / 0.42
P(B | A) = [0.12] / 0.42
P(B | A) = 0.29
Answer: Given that a student received a bid from Delta Apple Pi, the probability they did not wear a hat is 0.29.
With probability trees, it's hard to work backwards with conditional probabilities. The Bayes Theorem enables you to do that.