Actually, I like that metaphor! Let me work this out:
The bookie would see heads and tails with equal probability. However, the bookie would also sees twice as many bets when tails comes up. In order to make the vig zero, the bookie should pay out as much as comes in for whichever bet comes up, and that works out to 1:2 on heads and 2:1 on tails! Thus, the bookie sets the probability for Beauty at 1⁄3.
Actually, I like that metaphor! Let me work this out:
The bookie would see heads and tails with equal probability. However, the bookie would also sees twice as many bets when tails comes up. In order to make the vig zero, the bookie should pay out as much as comes in for whichever bet comes up, and that works out to 1:2 on heads and 2:1 on tails! Thus, the bookie sets the probability for Beauty at 1⁄3.