In keeping with the theme of computer-oriented techniques, we now show how accurate estimates of certain game-related probabilities can be achieved by enumeration of all possible cases and the appropriate weighting of each case. In the following example, we wish to evaluate the relative strength of a given Hold'em hand, in terms of the probability of currently having the best hand among active players.
Suppose our starting hand is K
-Q and
the flop is K
-7-2
.
Since there are 47 unknown cards in the remaining stock, there are only
1081 possible combinations of hands that our opponents might hold. We can
count how many of these are better, equal, or worse than our hand, and
determine where our hand is ranked against a random hand. It turns out
that 42 combinations are better, 6 are equal and 1033 are worse, which
corresponds to a percentile ranking of 1036/1081 = 0.958, or about a
96% chance that our hand will be better than a random opposing hand.
However, this is only a crude measure of hand strength because in
actuality, not all opposing hands are equally likely. For example, a hand
like A-A would certainly be played to this
stage of the game, but it is unlikely that a reasonable opponent would
have called a first round bet with 7-2
.
Since there are only 169 types of starting hands in Hold'em, each of the
1081 possible cases can be multiplied by an empirically determined
estimate of the probability that a hand would have been played to the
current point of the deal.
One implementation of this idea produces a probabilistic distribution of
about 23-6-346 better-tied-worse hands, yielding an actual strength of
0.931. The fact that the true strength is lower than the simple ranking
indicates that those hands which are currently superior to ours are, in
general, slightly more likely to be played by a rational opponent than a
random hand. In the absence of other information, we can approximate the
probability of having the best hand against two opponents to be 
, which is certainly strong enough to bet. This refined technique
for hand evaluation turns out to be surprisingly accurate and robust, even
when using rather simplistic estimates for each subcase.