The First Student

• Suppose he sees a red marble.

• If each marble is equally likely, the probability that a red marble is observed from "red"is 2/3 = P[B|A].

• We want P[A|B], however. We can get it from Bayes' Rule, but we need to know P[A] and P[B].

  • P[A] = 1/2, P[B] = 1/2.
  • We assume P[A] = 1/2. (In an experiment we could induce it by flipping a coin.)
  • Given the urns are equally likely, any of six marbles, 3 red, 3 blue, is equally likely, so P[B] = 1/2.

• Then:

  P[A|B] = P[B|A] (P[A] / P[B]) = (2/3) ((1/2)/(1/2)) = 2/3
  

  and the student who sees a red marble should bet on "red".

• Common mistake: substituting P[A|B] for P[B|A] without correcting for (possible) difference between P[A] and P[B].

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