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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