Probability Analysis for being dealt various
lowball (low ball) Poker hands including misc. wild cards
5 card lowball Poker hands with misc. wild cards
6 card lowball Poker hands with misc. wild cards
7 card lowball Poker hands with misc. wild cards
8 card, 9 card, and 10 card lowball Poker hands with misc. wild
Lowball (Low Ball) Poker Probabilities
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.
Hand Number of
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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