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
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.
- 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
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
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
- 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 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
- 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
- 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.
straight flushes with
- 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
- Straight & flush.
- This is a hand of the form
precisely 2 of the cards from ranks x,x+1,x+2 are in the same suit as yand
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
- 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
choices leaving 204. Altogether there are
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
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
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
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.
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.
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
|Type of Hand
||Number of Hands
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