Mambo Poker

Brian Alspach

11 January 2000

Abstract:

We determine the probabilities of the hands for mambo poker.

In mambo poker one uses the best 3-card hand which can be formed from 4 cards. We are going to count the numbers of ways of achieving the possible hands. Since the hands are based on 3 cards, when we use the word straight or flush throughout this file, we automatically mean 3-card straights and 3-card flushes, respectively. Any deviations from this will be named explicitly.

Since the game is played high-low, we examine both high and low hands. We consider high hands first. The total the number of possible 4-card hands is given by

$\begin{displaymath}{{52}\choose{4}}= 270,725.\end{displaymath}$

In order to avoid double counting certain hands, we shall mention a variety of 4-card possibilities and decide later how the hands should be valued. For example, there are 4-card hands containing both a straight and a flush. We shall distinguish them initially in order to make certain, for example, that straights really beat flushes.

The initial step is to determine all the types of hands to be counted. They are 4-of-a-kind, 4-card straight flush, 4-card flush containing a straight flush, 4-card flush not containing a straight flush, 4-card straight containing a straight flush, 4-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, 2 pairs, 1 pair and high card.

4-of-a-kind.: There are 13 possible ranks for the quartet and precisely 1 4-of-a-kind of each rank. Thus, there are 13 4-of-a-kind hands.
4-card straight flush.: Any card of rank ace through jack may begin a 4-card straight flush. Thus, there are 44 4-card straight flushes
4-card flush containing a straight flush.: There are 8 straight flushes beginning with an A or a Q. Any of 9 cards can be added to each to produce a flush which is not a 4-card straight flush. There are 40 straight flushes remaining and any of 8 cards can be added to each. This gives us $(8\cdot 9) + (40\cdot 8) = 392$ 4-card flushes containing staright flushes.
4-card flush not containing a straight flush.: We employ a technique in this case which will be used in several other cases leading us to explain it now in detail. There are ${{13}\choose{4}}= 715$ sets $\{x,y,z,w\}$ of 4 distinct ranks one can choose from 13 ranks. Of these, 11 have the form $\{x,x+1,x+2,x+3\}$ . Another 98 have the form $\{x,x+1,x+2,y\}$ , where y is neither x-1 nor x+3, for if x = A or x = Q, there are 9 choices for y and if x lies between 2 and J, inclusive, there are 8 choices for y. So removing these 109 sets of ranks leaves 606 sets $\{x,y,z,w\}$ of ranks which contain no straight or 4-card straight possibilities. Thus, for each of these 606 sets of ranks, there are 4 choices of suits giving us 2,424 4-card flushes containing neither a straight flush nor a 4-card straight flush.
4-card straight containing a straight flush.: There are 8 straight flushes beginning with A or Q. We can add any of 3 cards to each of them to obtain a 4-card straight containing a straight flush. To the remaining 40 straight flushes we may add any of 6 cards. This produces $(8\cdot 3) + (40\cdot 6) = 264$ 4-card straights containing a straight flush.
4-card straight not containing a straight flush.: There are 11 sets $\{x,x+1,x+2,x+3\}$ of ranks corresponding to 4-card straights. There are 4⁴ = 256 choices of the 4 cards, but some choices correspond to hands already counted: All in the same suit is a 4-card straight flush, and either the first 3 or the last 3 in the same suit gives a straight flush. There are 4 choices for the former and 24 choices for the latter. Removing these 28 hands already counted gives $11\cdot 228 = 2,508$ such hands.
Straight flush with a pair.: There are 48 straight flushes and any of 9 cards producing a pair as well. This gives $9\cdot 48 = 432$ straight flushes which also contain a pair. Notice the hand also contains a straight.
Straight flush without a pair.: If a straight flush begins with an ace or queen, any of 27 cards may be added without forming any hand which already has been counted. If a straight flush begins with any other card, one may add any of 24 cards without creating a hand already counted. This produces $(8\cdot 27) + (40\cdot 24) = 1,176$ straight flushes with no pairs.
3-of-a-kind.: There are 13 choices for the rank of the 3-of-a-kind, 4 choices for the 3 cards of the chosen rank, and the remaining card may be any of 48 cards. This yields $13\cdot 4\cdot 48 = 2,496$ 3-of-a-kind hands.
Straight & flush.: This is a hand of the form x,x+1,x+2,y, where precisely 2 of the cards from ranks x,x+1,x+2 are in the same suit as yand $y\not\in\{x-1,x,x+1,x+2,x+3\}$ . We saw above that there are 98 such sets of 4 ranks. There are 4 choices for the suit of y, 3 choices for the other 2 cards of the same suit, and 3 choices for the other suit. This gives us $98\cdot 4\cdot 3\cdot 3 = 3,528$ such hands.
Straight without a pair.: A straight without a pair has the form x,x+1,x+2,y, and we saw earlier that there are 98 such rank sets not allowing a 4-card straight. For each set of ranks, there are 4⁴ = 256 choices for the cards, but we must exclude 3 or 4 from the same suit in order to eliminate flushes. This eliminates $4 + (4\cdot 4\cdot 3) = 52$ choices leaving 204. Altogether there are $98\cdot 204 = 19,992$ straights without a pair.
Straight with a pair.: The set of ranks for this type of hand is $\{x,x+1,x+2\}$ . There are 12 such sets. There are 3 choices for which rank is paired and 6 choices for the pair of the chosen rank. The suits for the remaining 2 ranks cannot both agree with one of the suits of the pair or we would have a straight flush. Hence, there are 16-2 = 14 choices for the other 2 cards. This gives us $12\cdot 3\cdot 6\cdot 14 = 3,024$ straights with a pair.
Flush without a pair.: As we saw in an earlier case, there are 606 sets $\{x,y,z,w\}$ of ranks not admitting 4-card straights or straights. Since 4-card flushes have been counted already, we choose 3 of the ranks to be suited in 4 ways, there are 4 choices for the suit, and 3 choices for the remaining suit. We obtain $606\cdot 4\cdot 4\cdot 3 = 29,088$ flushes without a pair.
Flush with a pair.: There are ${{13}\choose{3}} - 12 = 274$ sets $\{x,y,z\}$ of ranks not allowing a straight. There are 4 choices for the suit of the flush, 9 choices for the card which pairs one of the flush cards and, thus, $274\cdot 4\cdot 9 = 9,864$ flushes with a pair.
2 pairs.: There are ${{13}\choose{2}} = 78$ choices for the ranks of the 2 pairs and 6 choices for each of the pairs. This yields $78\cdot 36 = 2,808$ hands of 2 pairs.
1 pair.: There are, as seen above, 274 sets $\{x,y,z\}$ of ranks not allowing a straight. There are 3 choices for the rank of the pair, 6 choices for the pair and 14 choices for the remaining 2 cards since we cannot allow both of the remaining 2 cards to be in the same suit as either suit of the pair as this would produce a flush. Therefore, there are $274\cdot 3\cdot 6\cdot 14 = 69,048$ hands with a single pair.
High card.: There are 606 sets $\{x,y,z,w\}$ of ranks not allowing a straight or 4-card straight. We also must eliminate flushes and in an earlier case saw that we are left with 204 choices for each set of ranks. This produces $606\cdot 204 = 123,624$ hands of this type.

If we add all the types of hands together we obtain 270,725 as we should. This tells us the above breakdown of all the 4-card hands is a partition of the set of all 4-card hands so there is no duplication.

There are 5 types of hands containing straight flushes. In spite of that, the sum of these types is smaller than any other kind of hand. We obtain 44 + 392 + 264 + 432 + 1,176 = 2,308 hands with a straight flush. There are 2 types of hands which contain 3-of-a-kind. Thus, there are 2,509 3-of-a-kind hands. We see that there is not a big distinction between these two premium hands.

The most interesting comparison is between straights and flushes. Let us see what happens if we rank a straight higher. This means every possible hand containing a straight will be counted as a straight. Doing so gives us 2,508 + 3,528 + 3,024 + 19,992 = 29,052 straights. The number of flushes becomes 2,424 + 9,864 + 29,088 = 41,376 and we see it is correct to rank straights as the better hand. In fact, a straight is considerably stronger as can be seen from the numbers.

The number of pairs is 71,856 since 2 types of hands will count as pairs. The number of high card hands is 123,624. Below is a table encapsulating the information.

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

We now examine the possible number of low hands given the rule the player must make a 6 or better to qualify for low. There are only two possibilities for a low hand. Either a player has a hand with 4 distinct ranks 3 of which are 6 or below, or the player has 3 distinct ranks all of which are 6 or below.

4 distinct low cards.: There are ${{6}\choose{4}} = 15$ possible choices for the 4 distinct low ranks. There are 4 choices for each of the ranks giving $15\cdot 4^4 = 3,840$ possible low hands of this type.
3 distinct low cards and 1 big card.: There are ${{6}\choose{3}} = 20$ choices for the 3 low ranks, there are 4 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.
3 distinct low cards and a pair.: There are 20 choices for the 3 low ranks as in the previous case, there are 3 choices for the rank of the pair, there are 6 choices for the pair, and there are 4 choices for each of the remaining cards. This yields $20\cdot 3\cdot 6\cdot 4^2 = 5,760$ .

We see there are only 45,440 low hands which means the probability of achieving a low is only

$\begin{displaymath}\frac{45,440}{270,725} = .1678\end{displaymath}$

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last updated 11 January 2000