**Brian Alspach**

**9 October 1998**

We determine the probability of a low board occurring for a board with 5 cards
from a standard deck of 52 cards.

In a game such as Omaha or hold 'em where 5 cards are displayed in the middle
for everyone to use in her or his hand, we refer to the 5 cards in the middle
as the *board*. When playing Omaha high-low, in order to qualify for low
a hand must achieve an 8-low or better. Thus, a low may occur only if the
board has at least 3 distinct ranks chosen from
.
We
refer to such a board as a *low board*.
Here we determine the probability of a low board occurring when the board is
dealt from a standard 52-card deck. We do so by counting the total number of
possible low boards and dividing by the total number of possible boards.
The total number of possible boards is

We first break the possible boards into six types depending on the number of big cards, denoted H, and the number of small cards, denoted L.

**Type A: Boards of the form H,H,H,H,H.**- Since there are 20 big cards
altogether, there are

boards of type A. **Type B: Boards of the form H,H,H,H,L.**- We are choosing 4 cards from the
20 big cards and 1 card from the 32 low cards. Thus, there are

boards of type B. **Type C: Boards of the form H,H,H,L,L.**- We are choosing 3 cards from the
20 big cards and 2 cards from the 32 small cards. Then there are

boards of type C. **Type D: Boards of the form H,H,L,L,L.**- Now we are choosing 2 cards from
the 20 big cards and 3 cards from the 32 low cards. This can be done in

ways. This is the number of boards of type D. **Type E: Boards of the form H,L,L,L,L.**- We are choosing one card from the
20 big cards and 4 cards from the 32 low cards. This gives us

boards of type E. **Type F: Boards of the form L,L,L,L,L.**- There are

boards of type F.

Boards of types A, B, or C are not low boards because they have at most two low
cards. We now break the boards of the three remaining types into subtypes, some
of which are low and some of which are not.

**D1.**- The three low cards are of distinct ranks. There are 3 choices for the
ranks from 8 ranks, there are 4 ways of choosing each of the low cards of a given
rank, and we are choosing 2 big cards from 20. This yields

boards of type D1. **D2.**- The three low cards contain a pair. There are 8 choices for the rank
to be paired, 6 ways of choosing a pair of that rank, 7 ways to choose the rank
which is not paired, 4 choices for a card of that rank, and a choice of 2 cards
from the 20 big cards. This yields

boards of type D2. **D3.**- The three low cards form 3-of-a-kind (trips). There are 8 choices for
the rank of the trips, 4 choices for the trips of that rank, and 2 choices for the
cards from the 20 big cards. This yields

boards of type D3.

Next we consider boards of type E.

**E1.**- The four low cards have distinct ranks. There are 4 choices for the ranks
from 8 ranks, there are 4 choices of card for each chosen low rank, there are 20
choices for the big card. This gives us

boards of type E1. **E2.**- There is one pair amongst the four low cards. There are 8 choices for
the rank of the pair, there are 6 choices of pairs of the given rank, there are
2 ranks to be chosen from the remaining 7 low ranks, there are 4 cards for each of
the other 2 low ranks, and there are 20 choices for the big card. Altogether this
gives

boards of type E2. **E3.**- The four low cards contain trips. There are 8 choices for the rank of
the trips, there are 4 choices for the trips, there are 28 choices for the remaining
low card, and there are 20 choices for the big card. This produces

boards of type E3. **E4.**- The four low cards form two pairs. There are 2 choices of ranks from 8
for the two pairs, each pair can be chosen in 6 ways, and there are 20 choices for the
big card. Multiplying yields

boards of type E4. **E5.**- The four low cards form 4-of-a-kind (quads). There are precisely 8 choices for the quads, and 20 choices for the big card. This yields boards of type E5.

Finally we consider boards of type F.

**F1.**- The five low cards have distinct ranks. There are 5 choices for the ranks
from 8 possible ranks, and for each rank there are 4 choices for cards. This produces

boards of type F1. **F2.**- There is a single pair among the 5 low cards. There are 8 choices for the
rank that is paired, there are 6 choices for the pair, there are 3 choices for ranks
among the remaining 7 ranks, and there are 4 choices for each of the latter 3 ranks.
We obtain

boards of type F2. **F3.**- There are trips occurring and two other low cards of different ranks.
There are 8 ways of choosing the rank of the trips, there are 4 ways of choosing
trips, the other 2 ranks are chosen from 7 ranks, and there are 4 choices for each
of the other two ranks. This yields

boards of type F3. **F4.**- There are 2 pairs amongst the 5 low cards. There are 2 choices from 8
for the ranks of the 2 pairs, there are 6 ways of choosing each pair, and there
are 24 choices for the remaining card. This yields

boards of type F4. **F5.**- The five low cards form a full house. There are 8 choices for the rank
of the trips, 7 choices for the rank of the pair, 4 choices for the trips of the
chosen rank, and 6 choices for the pair of the chosen rank. This produces

boards of type F5. **F6.**- There are quads occurring amongst the 5 low cards. There are 8 ways of choosing quads and 28 ways of choosing the remaining card. This produces boards of type F6.

The low boards are types D1, E1, E2, F1, F2, F3, and F4. Adding the numbers of
boards of these types yields 1,561,728 low boards. Dividing this number by
2,598,960, the total number of boards, gives us 0.601 as the probability of a low
board occurring for a standard 52-card deck. This implies the probability of
0.399 for no low being possible.

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