Pai Gow Poker

Brian Alspach

Poker Digest, Vol. 5, No. 7, March 22 - April 4, 2002

This article is a result of a recent correspondence with R. Timblin about Fortune pai gow poker and two possible side bets. Pai gow poker is played with a standard 52-card deck plus one joker. Players are dealt seven cards. All hands under discussion are standard five-card hands unless specifically mentioned otherwise. The joker may be used as an ace or as a wild card for straights and flushes.

The side bet under discussion here costs the player $1. The player receives the following payouts for the listed hands:

straight - pays $2;
three-of-a-kind - pays $3;
flush - pays $4;
full house - pays $5;
four-of-a-kind - pays $25;
straight flush - pays $50;
royal flush - pays $150;
five aces - pays $400;
seven-card straight flush with joker - pays $1,000;
royal flush with royal match - pays $2,000;
seven-card straight flush without joker - pays $8,000.

Let's determine the house edge for this particular side bet. In order to do this we need to know the probabilities for the occurrence of each of these hands.

The total number of seven-card hands from the standard deck augmented with a joker is $C(53,7) = 154,143,080$. The method employed is similar to what was done in ``Bring On The Jokers'' (Poker Digest, Vol. 4, No. 18).

The next table gives the numbers of types of hands for seven cards dealt from a standard 52-card deck.

hand number
straight flush 41,584
quads 224,848
full house 3,473,184
flush 4,047,644
straight 6,180,020
three-of-a-kind 6,461,620
two pair 31,433,400
pair 58,627,800
high card 23,294,460

The preceding table covers seven-card hands that do not contain a joker. In order to include the hands containing a joker, we take all possible six-card poker hands and see what happ[ens to them upon adding a joker.

There may be considerable migration of hands when a joker is present. The next table is a table of six-card poker hands. They are ranked according to the best five-card hand they contain.

hand number
straight flush 1,844
quads 14,664
full house 165,984
flush 205,792
straight 361,620
three-of-a-kind 732,160
two pair 2,532,816
pair 9,730,740
high card 6,612,900

When I say there may be considerable migration of six-card hands upon adding joker, consider six-card hands containing two pairs. A hand such as A-A-2-3-3-4 may become a full house or a straight flush depending on the suit distribution, whereas, 3-4-4-6-6-7 becomes either a straight or a straight flush upon adding a joker.

After seeing what happens to six-card hands upon adding a joker, the following table contains the numbers of the pertinent hands with seven cards. One interesting feature of counting these hands is that 3-of-a-kind is a better hand than a straight as far as the bonus payout is concerned. This is why the numbers for 3-of-a-kind and straight hands do not agree with the numbers for these hands when counting 7-card hands for a deck with a joker added.

hand number
7-card straight flush, no joker 32
royal flush with royal match 72
7-card straight flush with joker 196
five aces 1,128
royal flush 26,020
straight flush 184,644
quads 307,472
full house 4,188,528
flush 6,172,088
three-of-a-kind 7,672,500
straight 11,034,204
two pair 35,553,816
pair 64,221,960
high card 24,787,380

Return to the list of payouts given above for the bonus bet. Multiply the profit for each winning hand times the number of them in the preceding table; sum these numbers; subtract the number of hands for which the $1 is lost; and divide by the total number of seven-card hands. This gives an expectation of -.078, which is better than many available side bets.

The house edge for the bonus bet will vary according to the payout schedule in force. If you play this game and find an alternate payout schedule, you may use the numbers in the preceding table to determine the house edge following the above procedure. But don't expect much improvement.

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