Likelyhood of a good hand after you

[BradleyT]

Let's say we're at a 6 player table and all players at the table will play a hand that's in the top 20% (whichever ranking chart you like). It get's folded around to the small blind who also does not have a 20% hand. Since we know that 5 players did not have a 20% hand does the likelyhood of the big blind having a 20% hand increase much?

This is a common situation in SnGs where it's profitable to push your SB with nearly any two cards into the big blind because you estimate they'll only call with the top 20% of hands - yet it seems like they do have a top 20% hand a lot more than 20% of the time. We had a thread in the STTF forum about this but nobody really knew any mathematical answers about how folded cards increase the likelyhood of a blind having better cards.

Is the first part hard to figure out?

[pzhon]

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]

[BradleyT]

Thanks for the numbers pzhon.

Looks like there's an article in this months Internet Magazine on this very topic!