Thread: over cards
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Old 11-26-2005, 10:59 AM
AaronBrown AaronBrown is offline
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Join Date: May 2005
Location: New York
Posts: 505
Default Re: over cards

The number of players doesn't matter. Your figures for the turn and river are unconditional, that is they are the probability of getting an overcard on the turn or river regardless of whether or not you got one on the flop.

The unconditional probabilities are easy to compute. For each potential overcard rank, there is an 8% chance of an overcard on each board card dealt. So with KK there is one overcard rank (Ace). There is an 8% chance of getting an Ace on any specific board card. With QQ and two overcard ranks (Ace and King) there is a 16% chance. With 88 and six overcard ranks, there is a 48% chance.

With AA, it's not a bad approximation to multiply 8% by 3 to get the probability of flopping an overcard, and by 2 to get the probability of getting an overcard on the turn or river, regardless of whether or not you got one on the flop. But as the probabilities go up, the approximation gets worse. And even for Aces, multiplying 8% by 5 to get a 40% probability of an overcard somewhere on the board is not accurate, the correct answer is 35%.

The problem is the cases where you get more than one overcard. You have an 8% chance of getting an Ace on the turn, and an 8% chance of getting one on the river. The chance of getting Aces on both the turn and river is 0.5%, so the probability of getting at least one Ace is 8% + 8% - 0.5% = 15.5% (not the 17% in your table).

Here are the correct values for (in order) the chance of flopping an overcard, the chance of getting an overcard on the turn if you did not get one on the flop, the chance of getting an overcard on the river if you did not get one by the turn, and the chance of any overcard on the board.

KK 23% 7% 6% 35%
QQ 41% 10% 8% 60%
JJ 57% 11% 8% 76%
TT 69% 10% 7% 87%
99 79% 9% 5% 93%
88 83% 7% 9% 97%
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