**Poker Digest Vol. 2, No. 19, September 10 - 23, 1999**

In this last article in the series dealing with inclusion-exclusion,
we are going to examine a different setting for applying it. We
presented the principle of inclusion-exclusion in terms of counting
numbers of objects having certain properties. If we look at the formula
in Part I of this series and divide every term by *A*, the total number of
objects, we get terms of the form *A*(*i*)/*A*. These numbers are nothing
more than probabilities. That is, there is a corresponding
inclusion-exclusion
formula for probabilities. This can be advantageous because the computations
involved for determining the probabilities may be less complex than doing
the counting directly. Let *P*(*i*) denote the sum of the probabilities
of a set of *i* properties holding summed over the collection of all
subsets of *i* properties, that is,
*P*(*i*)=*A*(*i*)/*A*. Then the probability
*P* that none of the properties hold is given by

where

We now illustrate by looking at two examples involving calculations done
in earlier articles. Suppose we have a player in Omaha high-low holding
A-2-H-H, where H denotes a 9-10-J-Q or K, the board has three low cards
of distinct ranks,
but no ace or deuce, and we want to compute the probability no one else
has an A-2. We first compute the probability player *j* has an A-2.
Player *j* can hold A-A-A-2, A-A-2-2, A-A-2-x, A-2-2-2, A-2-2-x, or
A-2-x-y. There are three choices for A-A-A-2, nine choices for A-A-2-2,
choices for A-A-2-x and
so on. Altogether there are 6,675 choices for these hands. There are
*C*(43,4) = 123,410 ways of dealing four cards to player *j*. Thus, the
probability player *j* receives an A-2 is 6675/123410. Since there are
nine other players, we must multiply the preceding probability by nine
to obtain *P*(*i*). Doing so yields
*P*(1) = .48679.

We next compute *P*(2). We count the number of ways players *i* and *j*both receive an A-2 in their hands. There are 15,980,004 ways for this
to happen as there are only nine patterns for the two hands. We multiply
this by
*C*(9,2) = 36 because that is the total number of ways of
choosing two players from nine. We then divide by
*C*(43,4)*C*(39,4) which
is the number of ways players *i* and *j* can be dealt four cards apiece.
This gives
*P*(2) = .05667.

Finally, we do the same for three particular players being dealt A,2.
This is particularly easy because there is only one pattern for the three
players to receive an A-2 apiece. There are 7,532,300,160 deals. This
is then multiplied by *C*(9,3) and divided by
*C*(43,4)*C*(39,4)*C*(35,4)which produces .001190. Note that this agrees, as it should, with the
figure .0012 obtained in *Low Board Blues: Reprise and Coda*.
We then use the inclusion-exclusion formula to obtain

which agrees with the figure given in the article cited in the preceding sentence. However, you will note the result has been obtained with considerably less computation. Also, we note that William Chen, who just completed his Ph.D. in mathematics at UC Berkeley (congratulations William!), did a similar calculation using inclusion-exclusion to obtain the probabilities for other people being dealt A,2 given no cards have appeared in the board. He posted his calculations on RGP and they corroborated the figures given in the article cited above.

Let's now consider flushes in hold'em. Suppose we have a player who has
two hearts in her hand with three hearts in the board. She wants to know
the probability no one else has a flush. Again we assume there are nine
other players. Since there are eight hearts unaccounted for, there can
be at most four other players also holding two hearts in their hands.
The probability player *i* has two hearts is 28/990 because there are
*C*(8,2) = 28 ways for player *i* to have two hearts and
*C*(45,2) = 990possible hands for player *i*. Multiplying by nine and simplifying gives
*P*(1) = 14/55.

For players *i* and *j*, there are
*C*(8,2)*C*(6,2) = 420 ways both can
receive hearts and
*C*(45,2)*C*(43,2) = 893,970 they can receive two hands.
We have to multiply by
*C*(9,2) = 36 obtaining
*P*(2) = 15120/893970 =
8/473.

In a similar manner we obtain
*P*(3) = 28/96965 and
*P*(4) =
14/23950355. Inclusion-exclusion then tells us the probability no one
else has a flush is

which agrees (of course!) with the figure of .762 published in

Bib Ladder then asked me (as many readers probably are asking as well), ``Why did you do these problems earlier using direct counting methods which required considerably more computation than what you did above?'' This is a good question and this is what I told Bib.

The basic counting techniques we started with are fundamental and must be included in any counting toolkit. We then saw how they could be used to handle a wide variety of counting problems. They are more or less fail-safe in that more sophisticated tools do not always work and sometimes lead people astray whereas the basic techniques usually can be made to work although sometimes at the cost of more labor. Now inclusion-exclusion is a powerful counting technique but it is applicable only for certain problems. Even at that one still uses basic counting techniques when applying inclusion-exclusion so you have to be able to use them. The point is that you now have a choice of more than one method for certain counting problems. In the future we shall be free to bring inclusion-exclusion into the fray if it is useful.

Home | Publications | Preprints | MITACS | Poker Digest | Poker Computations | Feedback

website by the Centre for Systems Science

last updated 27 August 2001