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Old 08-31-2007, 10:45 AM
pzhon pzhon is offline
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Join Date: Mar 2004
Posts: 4,515
Default Re: Likelyhood of a good hand after you

It's painfully tedious to work out the bunching effect exactly for a reasonable distribution. Not only does it affect the probability that you are up against a top 20% hand, it also affects the probability you win when you do get called by a particular hand.

If you assume the players view that top 20% as any ace and any pair (20.4%), a lousy range but one that is easier to work with than most, then you get almost no information from the fact that they are unpaired, so let's ignore it, and only use the fact that the players do not have an ace. If you have no ace, then there are 4 aces left among 40 cards, and the probability your opponent has an ace is 1-(36/40)*(35/39) = 5/26 ~ 19.2%, as opposed to the 15.5% after you only see that you don't have an ace. The probability of having a pair changes, but only a tiny amount, so the probability that your opponent has a hand in this 20% range increases by roughly 4%. This should overstate this part of the bunching effect for more reasonable ranges.

For a better range, and more accurate results, you can use a simulation. I used a range of 55+, A7+, and any two broadway cards (20.2%), and ignored the information from your hand. In a million deals, 308,776 times the first 5 players folded, and of those, the big blind had a hand in the range 68,366 times, or 22.1%.

Mathematica code:
<ul type="square">
top20 = {{3, 3}, {4, 4}, {5, 5}, {12, 5}, {6, 6}, {12, 6}, {7, 7}, {12, 7}, {8, 8}, {9, 8}, {10, 8}, {11, 8}, {12, 8}, {9, 9}, {10, 9}, {11, 9}, {12, 9}, {10, 10}, {11, 10}, {12, 10}, {11, 11}, {12, 11}, {12, 12}}

intop20[card1_, card2_] := MemberQ[top20, Sort[{Mod[card1, 13], Mod[card2, 13]}, Greater]]

&lt;&lt; DiscreteMath`Combinatorica`
Sum[{
deal = RandomPermutation[52];
topqlist = Table[intop20[deal[[k]], deal[[k + 1]]], {k, 1, 11, 2}];
retvec = {If[tempbool = topqlist[[1]] || topqlist[[2]] || topqlist[[3]] || topqlist[[4]] || topqlist[[5]], 0, 1],
If[tempbool || topqlist[[6]], 0, 1]}}[[1]],
{j, 1000000}]

Output: {308776, 240410}[/list]
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