Steve Brecher
November 28, 2005
Most players know that pre-flop position is important in Texas Holdem. The
earlier your position, the more players there are behind
you and, unless you hold pocket Aces, the bigger the
chance that one of them will have a starting hand better than
yours.
There is another aspect to position: It's better to
act after your opponent(s) rather than before. But for
this tip, I'm going to investigate the chances that a
player behind you will have a better starting hand.
There is no universal definition of what "better"
means when comparing Texas Holdem starting hands. For this
article, I needed some reasonable, quantifiable
criterion. So in the following, I'm assuming that one
hand is "better" than another if its showdown equity is
greater. A hand's showdown equity against another hand
is the average portion of the pot it will win across all
possible combinations of board cards. This is similar to
the percentages that TV poker programs display next to
player hands when the players are all-in. If you're
interested in investigating this for yourself, there are
several free computer programs and websites which
calculate the showdown equities of user-specified
competing hands.
For example, Ah 2d all-in pre-flop against Kc Qc
will, over all possible boards, win an average of 53.9%
of the pot. So the A-2 is the "better" hand against K-Q
suited by our definition. Obviously, it is not better
for all purposes; at a full table I'd usually open-raise
in early position with K-Q suited, but toss A-2 offsuit.
Given some specific hand category – such as K-Q
suited – we'll need to know the chance that a random
hand dealt from the remaining 50 cards will be "better."
This requires that we have a showdown equity calculation
for each of the 1,225 possible opposing hands and
tabulate against how many of them the K-Q suited has the
worse (less than 50%) equity. It turns out that 238 of
the 1,225 possible opponent hands are "better" in this
sense. So we say that the chance of a random hand being
better than K-Q suited is 238/1,225 or 19.4%;
conversely, the chance that a random hand will not be
better is 80.6%. This tabulation would be too tedious to
do by hand. For the example results below, I developed
some simple software to do the calculations.
Suppose that you are considering an opening bet
pre-flop. There are players yet to act behind you. I'll
denote the number of hands to play behind you as N. For
example, if you're on the button, then there are two
hands - the blinds - behind you, and N would be equal to
2. What is the probability that none of some number of
random hands will be better than yours? It is the chance
that one random hand will not be better than yours
multiplied by itself N-1 times, which is the same as
saying it's that probability raised to the Nth power.
For example, if there's a 40% chance that a random hand
won't be better (i.e., a 60% chance it will be better),
then the chance that none of three random hands will be
better is 40% x 40% x 40%, or 0.4 to the 3rd power,
which equals 0.064. Hence, the chance that at least one
of the three hands will be better is 1.0 - 0.064 or
0.936 or 94%.
I think the most interesting thing about these
numbers is the difference between earlier and later
positions. This is something to consider when you're
thinking of open-raising in early position.
Steve Brecher
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