Quote:FinsRuleI did 88 / 55 / JJ against 888 / 88 and the full house won.

If you make it 99 / 44 / QQ against 888 / 88 or 888 / 99 - it's going to be really close, but I still think it'll be slightly tilted to full house. But maybe 3 pair would do better when simulating every single 3 pair possibility, not just the average.- WOO is blocked on my work computer so I can't do the calcs.

My guess is the math is way too time consuming to be worth figuring out a purely academic question.

I know the Wizard has said that the Joker really makes Pai Gow Poker calculations more complicated.

The page with the math information is blocked, but the chat site is available? Workplace governance win!! :-)

I should report this to Human Resources.

Oh wait, that's me, and I can't do anything about it.

Quote:mipletUsing the average WoO Relative Strength ( 0.95 * (5-card * 2-card) - (1-five * 1-two)) for each of the 156 different full houses and 286 different 3 pairs.

Note that I edited my original post due to a braino :)

Did you take into account that each of the 286 3 pairs does NOT come up equally? That to me is the entire point of this academic discussion.... Just as the most simple example, 22,33,AA will come up far more often than 22,33,44.

Full Houses

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Average low hand is 19.1% of the way between 88 and 99

Average high hand is 53.7% of the way between 888xy and 999xy

Three Pairs

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Average low hand is 81.6% of the way between JJ and QQ

Average high hand is 21% of the way between 88XXy and 99XXy

The low hand has 91 possible ranks, ranging from unsuited 23 all the way up to a pair of Aces.

The high hand has fewer than 7,463 possible ranks (the standard 7,462 plus one more for Five Aces). Some 5-card hands are not possible, such as 23457 unsuited, but let's ignore these for now and go along with the 7,463.

With Three Pairs, the average low hand is 3.625 ranks higher than the Full House's average low hand, an improvement of 3.625/91 = 4% of the range of possible low hands.

With Full Houses, the high hand's average rank is 858 ranks higher than the Three Pair's average high hand, an improvement of 858/7463 = 11.5% of the range of possible high hands. This is before eliminating the impossible high hands, which if we do, the 7463 would become smaller and therefore the 11.5% would become larger.

Since the Full House boosts the average high hand by a greater percentage than the Three Pair boosts the average low hand, I think the Full House is the better starting hand.

QQ + (99 wirh 88 or less) averages is 0.7161

Miplet's random three-pairs is between these two.

EDITED here for other responses...

My calculation does take that into account, thats why there are significantly more AA high three pairs (950,400) than any other X-high three pair hands. Note that with only the KK high three pairs added in, that nearly HALF of all possibilities are accounted. This would leave the other half ranked as QQ high three pair or less. The Confusing (and understandably so) Issue is "The Relative Strength" of the hand. This proceeds from a different point of view entirely. Is JJ/TT/99 a better hand by Relative Strength than QQ/33/22 ? The answer is yes even though QQ high out-ranks JJ high. The other two pairs contribute to the overall Relative Strength.

Quote:JBLooking only at 7-card hands that contain exactly 3 pair or exactly a full house (hands with trips+two pair, four of a kind, and five of a kind were skipped), here are the results:

Full Houses

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Average low hand is 19.1% of the way between 88 and 99

Average high hand is 53.7% of the way between 888xy and 999xy

Three Pairs

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Average low hand is 81.6% of the way between JJ and QQ

Average high hand is 21% of the way between 88XXy and 99XXy

The low hand has 91 possible ranks, ranging from unsuited 23 all the way up to a pair of Aces.

The high hand has fewer than 7,463 possible ranks (the standard 7,462 plus one more for Five Aces). Some 5-card hands are not possible, such as 23457 unsuited, but let's ignore these for now and go along with the 7,463.

With Three Pairs, the average low hand is 3.625 ranks higher than the Full House's average low hand, an improvement of 3.625/91 = 4% of the range of possible low hands.

With Full Houses, the high hand's average rank is 858 ranks higher than the Three Pair's average high hand, an improvement of 858/7463 = 11.5% of the range of possible high hands. This is before eliminating the impossible high hands, which if we do, the 7463 would become smaller and therefore the 11.5% would become larger.

Since the Full House boosts the average high hand by a greater percentage than the Three Pair boosts the average low hand, I think the Full House is the better starting hand.

Thanks, JB. But your analysis implies that each of the hands comes up proportionally. We all know that that is not true. Simplest example... AK is very common, A2 is very rare. I would guess the added EV from changing from A2 to A3 is almost zero, but the added EV from changing from AK to 22 (as player) is HUGE. Am I correct?

Quote:SOOPOOThanks, JB. But your analysis implies that each of the hands comes up proportionally. We all know that that is not true. Simplest example... AK is very common, A2 is very rare. I would guess the added EV from changing from A2 to A3 is almost zero, but the added EV from changing from AK to 22 (as player) is HUGE. Am I correct?

There is some truth in that, and my answer is that I don't know. Pai Gow Poker analysis is difficult unless you pretend that the dealer and player hands are dealt from separate decks; when you do that, analysis is (relatively) quick and easy, albeit slightly inaccurate.

The ideal way to answer the original question would be to:

A) Determine what strategy the dealer (or opponent) is using

B) Who is banking

C) What does a winning bet pay (is commission prepaid, do they only charge $1 when betting $25, is there no commission, etc.)

D) Analyze all full house hands against all possible opponent hands using the above knowledge

E) Analyze all three pair hands against all possible opponent hands using the above knowledge

F) Compare the results

That could take days to do a thorough analysis, or a couple of hours if cheating (pretending that the two hands are dealt from separate decks). I don't consider my previous answer final unless something like the above has been done (which I'm sorry to say I will not be doing), but I will cautiously stand by it in the meantime.