Poker Goes Latin: Part I

Brian Alspach

Poker Digest Vol. 2, No. 2, January 15 - 28, 1999

A new poker game, called mambo stud, recently has been introduced and has garnered some attention with some people predicting a bright future for the game. The game is based on three-card poker hands. Each player has three cards of her own, one face up and the other two down, and a community card available for every player which, of necessity, also is face up. The player forms the best three-card hand from the four cards. The game is played high-low and there is a qualifying rule for low, namely, six or better.

Mambo stud and sousem have striking similarities when it comes to evaluating hands. The similarities arise from the fact that many four-card hands contain more than one three-card hand. Since we just finished three articles on counting sousem hands, it seems appropriate to take a look at mambo stud. We examine low hands first because they are much easier to count. This lets us save the best to last. There are only two possibilities for a low hand. Either a player has a hand with four distinct ranks at least three of which are six or below, or the player has three distinct ranks all of which are six or below.

Four distinct low cards.
There are C(6,4) = 15 possible choices for the four distinct low ranks. There are four choices for each of the ranks giving $15\cdot 4^4 = 3,840$ possible low hands of this type.
Three distinct low cards and one big card.
There are C(6,3) = 20 choices for the three low ranks, there are four choices for each of the cards, and there are 28 choices for the remaining card. This gives $20\cdot 4^3\cdot 28 = 35,840$ low hands of this type.
Three distinct low cards and a pair.
There are 20 choices for the three low ranks as in the previous case, there are three choices for the rank of the pair, there are six choices for the pair, and there are four choices for each of the remaining cards. This yields $20\cdot 3\cdot 6\cot
4^2 = 5,760$.

We see there are 45,440 low hands. The total number of four-card hands is C(52,4) = 270,725. This means the probability of achieving a low is only

\begin{displaymath}\frac{45,440}{270,725} = 0.168\end{displaymath}

so that the odds against getting a low are 5:1.

We now consider high hands. Since the hands are based on three cards, when we use the word straight or flush, we automatically mean three-card straights and three-card flushes, respectively. Any deviations from this will be named explicitly. The three-card hands are straight flush, three-of-a-kind, straight, flush, one pair, and high card. We wish to make the counting as simple as possible, but there are two pitfalls to avoid. One is the possibility of duplicate counting and the other is the existence of four-card hands containing more than one three-card hand. The four-card hands are four-of-a-kind, four-card straight flush, four-card flush containing a straight flush, four-card flush not containing a straight flush, four-card straight containing a straight flush, four-card straight not containing a straight flush, straight flush with a pair, straight flush without a pair, 3-of-a-kind, straight & flush, straight without a pair, straight with a pair, flush without a pair, flush with a pair, two pair, one pair and high card. Every four-card hand contains four different three-card hands. In a certain sense, this means a player has four choices as to what the hand is worth. In practice the choice is normally obvious. For example, if a hand contains a straight and a pair amongst other things, the player will treat the hand as a straight because a straight is ranked higher (see below). The point is that when determining the correct hand rankings, one must consider all the possibilities for various four-card hands and make certain the hand rankings are consistent with the likelihood of achieving the hand. That is, the higher a hand is ranked, the less likely it is to occur.

The following table shows the number of different types of hands in the second column. The third column contains the probabilities of achieving the various hands.

Type of Hand Number of Hands Probability
Straight flush 2,308 0.0085
3-of-a-kind 2,509 0.0093
Straight 29,052 0.107
Flush 41,376 0.153
One pair 71,856 0.265
High card 123,624 0.457

The number of high hands adds to 270,725 as it should. There are a few observations we can make from the table. First notice the two premium high hands, straight flush and three-of-a-kind, are roughly the same in number. However, there is a huge gap between straights and the two premium high hands. There are more than 11 times as many straights as there are three-of-a-kind hands. A flush is considerably weaker than a straight as there are more than 40 percent more flushes than straights. The latter fact may surprise a few people. The gaps from flush to a single pair, and from a pair to high card also are very big.

In the next article we shall discuss counting the high hands.

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