**Brian Alspach**

**Poker Digest Vol. 1, No. 4, September 11 - 14, 1998**

Woke up this mornin' Omaha on my mind

Woke up this mornin' Omaha on my mind

Don't wanna play no low hands

'Less I'm in the big blind

As I pulled into the parking lot of the local cardroom, I stopped
singing my rendition of the *Low Board Blues*.
When I entered the room, I saw Bib Ladder sitting by himself drinking
a cup of coffee so I asked if I could join him. He invited me to take
the chair opposite him.

``Actually, professor, I'm glad ya dropped in. I bought a computer three weeks ago and have discovered the poker newsgroup. They've been droppin' a lot of numbers about the chances of making low hands in Omaha high-low. Since I understood your earlier explanation about the number of three-card poker hands, I was hopin' you could show me how to get these numbers.''

``Sure, Bib. I'll retrieve my ubiquitous pad of paper and see what we can come up with. The counting is a little more sophisticated for this problem, but as long as you keep common sense and the basic principles I mentioned earlier in mind, we should be alright. Something else we shall develop is that the numbers change according to your viewpoint; more about that later.''

I continued, ``Let's assume that you are on the rail watching the
boards appear and you ask yourself what the chances are that a
low will not be possible. Let's first calculate the
total number of boards. We are choosing 5 cards from 52. As
we saw the last time we talked, this number is *C*(52,5). Evaluating
this yields
divided by
.
After doing the arithmetic, we
find that there are 2,598,960 possible boards.

The question now is how many of those boards preclude a low. The board
may contain five high cards, such as K-Q-J-10-9, or four high cards
and a low card, or three high cards
and two low cards, and so on. Since I am using the word *or*, we have
to add the numbers we get. If the board contains five high cards, then we
have chosen five cards from the 20 high cards in the deck.
This is
which is 15,504. Next, the board
may contain four high cards and one low card. The four high cards can be
chosen in
ways. The single low card can be any of 32 cards. We have to multiply
this by 4,845 because we are choosing 4 high cards AND one low card -
remember that *and* means multiply. Anyway,
we obtain 155,040 such boards. Is this alright so far?''

``Yeah, professor, I understand so far. Continue.''

``Next we want to count the number of boards with three high cards and two
low cards. It will be the product of the number of ways of choosing three
high cards from the 20 available and two low cards from the 32 available.
This is the product of *C*(20,3) and *C*(32,2). We easily calculate
that
*C*(20,3)= 1,140 and
*C*(32,2)= 496. Their product
is 565,440. All of the preceding boards preclude a low hand because
there are at most two low cards in the board. Things get a little stickier
now because three or more low cards will be in the board. Just keep the
common sense here and we shall prevail.

We now move to boards with two high cards and three low cards. The two high
cards can be chosen in
*C*(20,2)=190 ways.
The problem is that even though three low cards are present, a low may be
precluded because at least two of the low cards form a pair. So we have
to break this into subcases. If the three low cards form trips, this can
happen in
ways because there are eight possible ranks for
low cards and four ways of choosing trips. The other way of counterfeiting
a low is that exactly two of the low cards form a pair. There are eight
choices for the rank that is paired, seven choices for the rank that is
not paired, *C*(4,2)= 6 ways of choosing the pair and *C*(4,1)=4 ways
of choosing the card
that is not paired. This gives us
.
We
do not have to divide by anything in the last product because we have
counted any such scheme exactly once. Altogether we have
1,344+32=
1,376 ways that three low cards do not allow a low. So there are
boards with precisely three low cards which preclude
a low. Are you still with me, Bib?''

``Let me think for a minute,'' he replied. After a short silence he nodded for me to continue.

``The next situation is four low cards on board that preclude a low. How can that happen?''

Bib immediately replied, ``Well, if there's quads, trips or two pair there can't be a low.''

``That's right. Since you said *or* that means we have to add those
three counts together. There are precisely eight ways to get quads.
There are eight ranks to choose for the trips, seven
choices for the rank of the other low card, four ways of choosing trips
and four choices for the other card. That gives us
ways. If there are two pair, there are *C*(8,2)=28 ways to
choose the two ranks to be paired, and each pair can be chosen in six ways.
This gives
ways for two pair to occur. Adding them
together gives 1,912 ways that four low cards can occur and yet preclude
any low. For each of them we can choose any of 20 high cards so that
the number of boards with 4 low cards and no low possible is
.

The final situation is having five low cards and no possible low. Using the same reasoning you just demonstrated, Bib, we must have either four-of-a-kind or a full house on board. There are eight choices for the rank of the card in the quartet and seven choices for the rank of the remaining card. The latter card can then be any of four yielding boards with four-of-a-kind. For full houses, we have eight choices for the rank of the trips, seven choices for the rank of the pair, four choices for the trips and six choices for the pair. Multiplying them gives 1,344 such boards.

Now we add all of the above boards together to obtain 1,037,232 boards which do not allow a low. We obtain the probability that no low is possible by dividing that by the total number of boards which, as we saw earlier, is 2,598,960. This gives us a probability of 0.399. So roughly 60 percent of the time an observer on the rail will see that a low is possible.

Oh, I see my seat is open. Let's continue this conversation later, Bib. Why don't you see if you can determine the probability of a low being possible given that you hold exactly four high cards. That may get you singing the blues too.''

As I headed towards my seat, Bib gave me a quizzical look as he tried to understand what that last comment was about. Next issue: The Low Board Blues: Part II.

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