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Poker Articles |
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The Basics of the
Mathematics of Poker
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Do you have questions
about probability in
poker games? “I went
all-in with pocket kings
before the flop, but
somebody else had aces!
What are the odds of
that?” “What are the
chances of winning with
AK versus 88?” |
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If you've ever wondered
how to answer these
questions, you require
some basic poker math. |
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Unknown Cards |
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One concept basic to all
poker math is that of
unknown cards. Although
cards have been dealt to
other players, we do not
know what any of them
are, so they do not
affect the probability
calculations we need to
perform. If five players
at a table receive AK,
QJ, T9, 87, and 64 of
hearts, and the flop has
two hearts, how would
each player calculate
their chance of filling
the flush? The answer is
that nobody can assume
less than nine outs,
because they have no way
of knowing what the
other dealt hands were. |
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Dealing Two Cards |
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In Texas holdem, each
player receives two hole
cards. There are 52
unknown cards when the
dealer sends out our
first card. Once we know
our first card, there
are 51 unknowns left in
the deck. We have an
equal probability of
getting any one of them,
so there are 52 x 51 =
2652 possible starting
hands. Since the order
we receive the cards
does not affect the
strength of the hand, we
further divide that
number by two to get the
result of 1326 possible
starting hands. |
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Since suits don't affect
the strength of the
hand, although having
two of the same suit
does, we can further
eliminate duplicate
starting hands. In this
calculation, 7-Clubs
4-Diamonds is the same
as 7-Spades 4-Hearts.
Now we reveal three
basic hand types: pocket
pairs (two cards of the
same rank), suited cards
(two cards of different
ranks but the same
suit), and unsuited
cards (rank and suit
both different). There
are 13 possible pocket
pairs, 78 suited
combinations, and 78
unsuited combinations,
for a total of 169
possible starting hands.
This number is much more
manageable than 1326. |
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For the suited
combinations, the first
card can be any of
thirteen ranks, and the
second is one of twelve.
13X12 = 156, and since
order of delivery
doesn't matter, we
divide by two, revealing
78 suited rank
combinations (AK, AQ...A2,
KQ, KJ...K2, QJ, and so
on). The same
calculation applies for
the unsuited cards. |
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Probability versus
Odds |
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Probability is a term
used to express the
chance that something
will happen. Odds are
the term used to express
the chance that
something will not
happen. If I pick one
card of 52, there is a
0.077 (7.7%) probability
that it will be a seven,
ignoring suits. The odds
of it being a seven are
12-1 against. |
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Getting Pocket Aces |
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One common poker math
question concerns the
probability of getting
dealt pocket aces. The
answer is that there are
six possible ways to get
two aces out of 52
cards: A - Spades A -
Hearts, A - Spades A -
Diamonds, A - Spades A -
Clubs, A - Diamonds A -
Clubs. There are 1326
possible two card
combinations, and six of
them are pocket aces, so
the probability of
getting aces is 6/1326 =
1/221 = 0.0045, or less
than one half of one per
cent. |
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Getting Dominated |
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If you are holding a
pocket pair, you know
the chance that someone
holds a pair higher than
yours. If you have KK,
there are (50x49)/2=1225
possibilities for the
pocket cards of any
opponent. Six of those
hands are pocket aces.
The probability that any
specific opponent has AA
is 6/1225 = 0.49%. For 8
opponents at a
nine-handed table, the
probability that any one
of them has AA is 1 -
(1219/1225)^8 = 0.0385,
or 3.85%. |
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If you have QQ, the
probability that one
opponent has AA or KK is
1- (1213/1225)^8 =
7.57%. |
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The study of poker
mathematics is a
fascinating subject if
you are positively
inclined towards math at
all. There are a million
probabilities that can
be calculated, and
studying odds away from
the table can improve
your play. |
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