Durango Bill's
Lowball (Low Ball) Poker Probabilities
Probability Analysis for being dealt various lowball
(low ball) Poker hands including misc. wild cards
Probabilities
for 5 card lowball Poker hands with misc. wild cards
Probabilities
for 6 card lowball Poker hands with misc. wild cards
Probabilities
for 7 card lowball Poker hands with misc. wild cards
Probabilities
for 8 card, 9 card, and 10 card lowball Poker hands with misc. wild
cards
There are several variations of “Low Ball”
Poker. The calculations presented here are for “Ace to
Five” (California Lowball) rules. For the Ace-to-Five game, you
select the five lowest cards from your hand that do not contain pairs.
Ignore straights and flushes. The “best” possible hand is a
five high - Ace, two, three, four, five. (Aces are always low.) If you
have to include a pair, it is legal, but it is a poor hand that will
lose to any hand that does not include a pair.
After a “5 high” hand, the next best hand is a
6 high. This is followed by a 7 high, etc. In case of a tie between two
hands, the tie is broken by the next highest card, etc.
If wild cards are included, they may be declared as the
lowest non-rank matching cards needed to fill out your five card hand.
The probabilities shown here are in sorted order (winning/best hands
are at the top). After displaying the table we will concentrate on how
to calculate these numbers.
Lowball
Hand Number of
Combinations Probability
--------------------------------------------------------
Five
high
1,024 0.00039400
Six
high
5,120 0.00197002
Seven
high
15,360 0.00591006
Eight
high
35,840 0.01379013
Nine
high
71,680 0.02758026
Ten
high
129,024 0.04964447
Jack
high
215,040 0.04964447
Queen
high
337,920 0.13002124
King
high
506,880 0.13002124
Pair or
worse
1,281,072 0.49291717
Total
2,598,960 1.00000000
= COMBIN(52,5)
The first calculation that must be made is to determine
the total possible lowball Poker hands. A Poker hand consists of 5
cards randomly drawn from a deck of 52 cards. Thus, the number of
combinations is COMBIN(52, 5) = 2,598,960. For each of the above
“Number of Combinations” we divide by this number to get
the
probability of being dealt any particular hand.
In order to find each “Number of Combinations”
entry in the above table, we first calculate how many different ways
there are of getting 5 different ranks, and then multiply by the number
of ways that the suits can be varied.
For the best possible hand, the highest rank is a
“Five”. The next four ranks must be chosen from the
remaining lower ranks (Ace, Two, Three, Four). The only way to do this
is to use each of these lower ranks exactly once. Technically, we are
choosing 4 combinations from the four available ranks. Combinatorially,
this is represented as COMBIN(4, 4) = 1. Thus the only ranks that can
be represented in a five-high hand are just 5, 4, 3, 2, Ace.
For each rank we can have one card from any of 4 suits.
(Clubs, Diamonds, Hearts, Spades.) The total suit combinations for the
5 ranks is thus 4 x 4 x 4 x 4 x 4 = 1024. This suit combination total
of 1,024 will apply to any rank combination that we might use. Thus the
total combinations for any “N” high hand will equal the
number of rank combinations times this 1,024 (the number of
suit combinations). For a “5 high” hand there is only one
rank combination. We multiple “1” by 1,024 to get 1,024
different ways that you can get a “5 high” hand.
When we get to a “6 high” hand, we have to
select the other 4 ranks from the available 5 lower ranks. The 5 lower
ranks are of course Ace, Two, Three, Four, Five. The number of ways
that this can be done is COMBIN(5, 4) = 5. (Select 4 ranks from the 5
that are available.) Finally we multiply the 5 possible rank
combinations by the 1,024 suit combinations to get the 5 x 1024 = 5,120
total combinations for a “6 high” hand.
Calculations for a “7 high” hand use the same
methodology. After using a “7” as a high card, we select 4
other ranks from the remaining 6 lower ranks. The 4 lower ranks can be
selected in COMBIN(6, 4) = 15 different ways. After multiplying 15 by
1024 we get the 15,360 different ways that you can get a “7
high” lowball hand. The remaining combinations for each
“N” hand are calculated in a similar way.
After you have calculated all possible “N”
high hands you can add the results together to get 1,317,888 possible
“N” high hands. This leaves 2,598,960 - 1,317,888 =
1,281,072 hands that have a pair or worse.
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