**Brian Alspach**

**11 January 2000**

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

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
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
sets
of
4 distinct ranks one can choose from 13 ranks. Of these, 11 have the form
.
Another 98 have the form
,
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 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
4-card straights containing a straight flush.
**4-card straight not containing a straight flush.**- There are 11 sets
of ranks corresponding to 4-card straights. There
are 4
^{4}= 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 such hands. **Straight flush with a pair.**- There are 48 straight flushes and any of 9 cards producing a pair as well. This gives 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
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
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*y*and . 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 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 4^{4}= 256 choices for the cards, but we must exclude 3 or 4 from the same suit in order to eliminate flushes. This eliminates choices leaving 204. Altogether there are straights without a pair. **Straight with a pair.**- The set of ranks for this type of hand is
.
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
straights with a pair.
**Flush without a pair.**- As we saw in an earlier case, there are 606
sets
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
flushes without a pair.
**Flush with a pair.**- There are
sets
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,
flushes with a pair.
**2 pairs.**- There are
choices for the ranks of
the 2 pairs and 6 choices for each of the pairs. This yields
hands of 2 pairs.
**1 pair.**- There are, as seen above, 274 sets
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
hands with a single pair.
**High card.**- There are 606 sets 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 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 possible choices for the 4 distinct low ranks. There are 4 choices for each of the ranks giving possible low hands of this type.
**3 distinct low cards and 1 big card.**- There are 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 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 .

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

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