Concept
Why is conditional probability not commutative?
Stephen Davies, Ph.D. Version 2.2.2 Through Discrete Mathematics A Cool Brisk Walk / Chapter 1
"Keep in mind, by the way, that unlike union and intersection, conditional probability is not commutative. In other words, Pr(X | Y) ≠ Pr(Y | X) in general. To take just one example, look again at the F and U sets from All-time Idol. Pr(F | U), as we already computed, is equal to 1 since if U has occurred, we automatically know that F has also occurred (there aren’t any underage contestants except females). But the reverse is certainly not true: just because we have a female winner doesn’t mean we have an underage winner, since the winner might be Carrie. Working it out, Pr(U | F) = Pr(U ∩ F) / Pr(F) = .4 / .5 = .8. Higher than Pr(U), but not 1."
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