Low Board Probability
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 $\{A,2,3,4,5,6,7,8\}$. 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 \begin{ thaibet168 }{{52}\choose{5}} = \frac{52!}{5!47!}= 2,598,960.\end{displaymath}
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 \begin{displaymath}{{20}\choose{5}}= \frac{20!}{5!15!}= 15,504\end{displaymath} 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
\begin{displaymath}32{{20}\choose {4}}=\frac{32\cdot 20!}{4!16!}= 155,040\end{displaymath}
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
\begin{displaymath}{{20}\choose{3}}{{32}\choose{2}}= 565,440\end{displaymath}
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 \begin{displaymath}{{20}\choose{2}}{{32}\choose{3}}= 942,400\end{displaymath}
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
\begin{displaymath}20{{32}\choose {4}}=\frac{20\cdot 32!}{4!28!}= 719,200\end{displaymath}
boards of type E.
Type F: Boards of the form L,L,L,L,L.
There are \begin{displaymath}{{32}\choose{5}}=\frac
{32!}{5!27!}= 201,376\end{displaymath} boards of type F.
Adding the numbers of boards of the six types yields 2,598,960 boards as it should.
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.
Web: https://thaibet168.school.blog/2021/10/13/low-board-probability/
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Public Last updated: 2021-10-17 10:01:28 AM
