Re: Running It Twice
Here is an example showing how EV is the same even without reshuffling. This is not a proof that this always works. I'll try to get that in another post.
Suppose you have KK, opponent has 55. Board is K952.
He wins 1/44 times. EV = 1/44*pot for opponent.
Suppose you run this three times without reshuffling:
He has 2 possibilities:
Lose all 3:
(43/44)*(42/43)*(41/42) = 41/44. EV = 0
Win 1, lose 2:
3/44 (1- 41/44).
Also: (1/44)+(43/44*1/43)+(43/44*42/43*1/42) = 1/44+1/44+1/44 = 3/44.
His EV = 3/44 * 1/3 pot = 1/44*pot.
Total EV for taking 3 cards is 1/44+0 = 1/44. Exactly the same as taking 1 card. But there is less variance since he wins 1/3 of the pot more often.
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