Bifalgorithm post · Brisk · Chapter 1

Why is conditional probability not commutative?

Why is conditional probability not commutative? (Brisk · Chapter 1) — "Keep in mind, by the way, that unlike union and intersection, conditional probability is…

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  • La probabilité conditionnelle n’est pas commutative.
  • Contrairement à l’union et à l’intersection, l’ordre des événements conditionnels compte.
  • En général, Pr(X | Y) ≠ Pr(Y | X).
  • Exemple : les ensembles F et U de All-time Idol.
  • Pr(F | U) = 1, car tout gagnant mineur est automatiquement une femme.
  • La réciproque n’est pas vraie : une gagnante peut être Carrie, qui n’est pas mineure.
  • Pr(U | F) = Pr(U ∩ F) / Pr(F) = .4 / .5 = .8.
  • Ainsi, Pr(U | F) est supérieure à Pr(U), mais elle n’est pas égale à 1.
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Why is conditional probability not commutative?

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Brisk

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"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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