Maths question
 

I've been wanting to learn how to determine how often a player with a particular range flops a particular pair on a particular flop. For example: How often would a player with a 10% range(according to pokerstove) flop exactly one pair of queens on a Q84 rainbow board (so his hand can't be better or worse than a pair of queens).

My way of working this out was to work out the total combinations of cards that make up a a 10% range. In this case is 118 combinations of cards considering that board. The total number of queen combinations is 30. Therefor 30/118 = 25.4% of the time that person should have exactly a pair of queens on that flop. Is this method correct?

Total combinations of cards::
Pairs:
6 AA, 6 KK, 3 QQ, 6 JJ, 6 TT, 6 99, 3 88 = 36 pairs

Ax:
16 AK, 12 AQ, 16 AJ, 4 AJs, 4 ATs, 4 A9s = 56 Ax

Kx:
12 KQ, 4 KJs, 4 KTs = 20 Kx

Qx: 3 QJs, 3 QTs = 6 Qx


Total combinations of cards = 118

Combinations of cards containing queens:

AQ = 12
KQ = 12
QJs = 3
QTs = 3

Total combinations of cards containing Q's = 30

Another method has been suggested by bryce of stoxpoker using pokerstove. In this method you would input that board(Q84 rainbow) on pokerstove and fill out the turn and river with blank cards like 22 or 23. Then you give yourself the best hand that loses to a pair of Q's, which would be JJ, and give your opponent a 10% range and see what your winning percentage would be. In this case it would be 52%. After that you give yourself the worst hand that beats a pair of Q's, which in this case would be KK, and work out your winning percentage. In this case the answer is 86%. Then you subtract answer 2 from answer 1 and that should give you the amount of times your opponent has exactly a pair of Q's. In this case our answer is roughly 34%.

Clearly this answer is different to my answer and a 9% difference is pretty large. Can anyone tell me where my method or the pokerstove method is going wrong?

Re: Maths question
 

Your method is definitely correct but it can be time consuming .

Here is one way of thinking about it by using an approximation.

A player with a top 10% range will be playing slightly more ace hands than kings hands and slightly more king hands than queen hands . We may say that they are about equal to make the problem easier for us . So the probability he has an A,K or Q are about equal . The probability he will be playing a pocket pair is ~ equal to the probability he would have an A , K or Q .

Therefore the probability your opponent flops exactly one pair of queens on a q-8-4 board is ~ equal to the probability he would flop one pair of kings on a k-8-4 board and ~ equal to the probability he would flop one pair of aces on a A-8-4 board . So if x denotes the probability he has an A,K,Q or a pocket pair , then we have that
x+x+x+x =1 and x=0.25 and we're done !

Re: Maths question
 

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Clearly this answer is different to my answer and a 9% difference is pretty large. Can anyone tell me where my method or the pokerstove method is going wrong?

[/ QUOTE ]
The PokerStove method Bryce mentioned on StoxPoker is very good for some situations, but not this one. The problem is that it is sort of like using a regular thermometer to measure the temperature of a drop of water. The initial temperature of the thermometer biases the reading. It's not as much of a problem if you are measuring the temperature of a larger quantity.

When you give yourself a testing hand in PokerStove, you block many combinations from among your opponent's holdings. When you give yourself JJ, you block 5/6 of the JJ combinations, as well as many AJ, KJs, and QJs hands. When you give yourself KK, you block KK, AK, KQ, KJs, etc. These don't have to balance out, and they don't here, primarily because having JJ blocks many more hands lower than one queen than hands with top pair or better.

The PokerStove method is more reliable when you are not using it on a small range like the top 10%. You will generally find smaller errors when you analyze a larger range like the top 50% of hands.

There ought to be a simple tool which tells you the answer without having to add dead cards, and some poker programs give that information.

Re: Maths question
 

Awesome, thanks for the great replies guys. Phzon which poker programs were you referencing that can do these calculations?

Re: Maths question
 

[ QUOTE ]

There ought to be a simple tool which tells you the answer without having to add dead cards, and some poker programs give that information.

[/ QUOTE ]

yea definitely.. this would be great. Do you know a tool that does this directly?