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#7
08-13-2002, 08:05 PM
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Now you have! The answer may surprise you

I have been chewing on this calculation. My final result is 1,215,008 or only 47% of all possible 5 card hands. Details are below.

I was surprised at first that this percentage is so low, but it makes sense when you realize that there are a high percentage of hands with one of two ranks that must improve, usually 2 or 3. Even a rapid application of my 6% rule gives a crude estimation that 60% of hands must improve, but that gives an overestimate.

I suppose there may be some value in knowing the total possible hands you can make on the river in holdem, or on 7th street in stud. Perhaps Mr. Sklansky already knows this result and can verify it. Otherwise, if other's want to site it in future works, I will hereby name it the BruceZ number. If other's make minor corrections to the following calculation, the result shall still be called the BruceZ number [img]/images/smile.gif[/img]

Here are the details:

5 card hands which must improve with 7 cards:

HIGH CARD: Any containing a 2 or 3 and the specific hands 45679, 45689.

8*44*40*36*32/4! + 16*44*40*36/3! - 8C(12,4) -

2(4)^5 + 8 + 2(4)^5 = 840,848

The first term is for hands without a pair that have a 2 or 3 but not both. The second term is for both a 2 and a 3. Then I have subtracted off flushes and straights, then added back the 8 straight flushes so they aren't subtracted twice, then added in the specific hands.

1-PAIR: Any containing the 3rd kicker from the two lowest unpaired denominations. For example, 4459x must improve for x = 2 or 3. 2259x must improve for x = 3 or 4. Also the following must improve: 44567, 44568, 45567, 45568, 45667, 45668, 45677, 45688.

13*6*8*40*36/2 + 13*6*16*40 + 8*6*4^3 = 502,272

The first term considers kickers from one of two distinct denominations, and the second considers kickers from both denominations. I've considered all the specific hands at once in the last term.

2-PAIR: Any with with both pairs higher than 22 and a kicker of 2 must improve. If both pairs are higher than 22 and the kicker is not 2, then it does not have to improve since drawing 22 will not improve it. 2233x improves for x=4,5,6. 2244x improves for x=3,5,6. 2255x improves for x=3,4,6. 2266x improves for x=3,4,5. Other 2-pair hands with 22 improve for kickers of 3 or 4.

4(12*6)(11*6) + 4*6*6*12 + 8*6*6*8 = 23,040

The first term is for 2xxyy. The second term is the 4 specific hands with 3 kickers, and the third is the 8 specific hands with 2 kickers.

3-OF-A-KIND: Any with 2nd kickers of 2 or 3. Also, 22245, 22256, 33345, 33356, 44456.

13*4*8*40 + 13*4*16 + 5*4*4*4 = 17,792

The first term is for kickers of 2 or 3 but not both, and the second is for kickers of both 2 and 3. The last term is for the 5 specific hands.

Straights, flushes, full houses, and quads, would not have to improve.

TOTAL hands that must improve = 1,383,952

Total hands on river = C(52,5) - 1,383,952 = 2,598,960 - 1,383,952 = 1,215,008.

1,215,008/2,589,960 = 47%.