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Lesson: 22
No-limit by the Numbers
Andy Bloch
August 15, 2005
I get asked a lot of poker strategy questions, from beginner to
advanced. Some are easy, but some involve the kind of math I can't
always do off the top of my head. When that happens, I rely on one
of a number of free tools to calculate the probability of winning
the hand.
Here's an example based on a hand posted on a website I run:
Our hero was playing at a small stakes No-Limit table online, with
$.25-$.50 blinds. At the start of the hand, he had $44. He was dealt
Ad-Td and raised to $2. Both blinds called. The flop was Kd-Jd-2c,
giving our hero a royal flush draw. The big blind bet $2, hero raised
$2 more, the next player called, and the big blind (with more chips
than our hero) re-raised all-in.
Should our hero call with his last $38? Let's assume the third
player will fold. If our hero were to call and win, he'd be up to
$94 (the $18 in the pot, plus his $38 and his opponent's $38). If
he wins the hand four times out of 10, on the average he'd have
$37.60 after the hand ($94 multiplied by four, and divided by 10).
In poker, it's the long run that matters, so he should only call
if his probability of winning is greater than 40%. Now he needs
to figure out the probability he'd win the hand.
The first step is to put his opponent on a range of hands. Sometimes,
you can figure out exactly what your opponent must have by the betting
or tells. Most of the time, you're left to guess a little. In this
situation, the other player probably has a very strong hand, but
there's a chance he's bluffing or even semi-bluffing.
The strongest hand our hero could be facing is three kings. He
has 11 outs to win the pot - every diamond but the 2d, and three
queens. But even if our hero makes his flush or straight, his opponent
could still win by making a full house or quads on the last card.
I could calculate the probability by hand, but I don't need to.
Instead, I head to the Internet and one of the many free poker
odds calculators, such as the one at twodimes.net. Enter "Kd
Jd 2c" in the box labeled "Board" and "Ad Td"
and "Ks Kc" under "Hands", and click submit.
The result says that Ad-Td wins under 34% of the time - less than
the 40+% that would make a call the right play. If our hero knows
that his opponent had three kings, he should fold. The probabilities
for the other possible three-of-a-kinds are the same.
But what if he's up against two pair - kings and jacks? Using the
poker calculator again, his probability of winning would be 44%.
That's enough to make calling correct. Our hero might also be against
other two pairs, which he'd beat a little less often (42%), or A-K
(46%). He might even already be ahead if he's against an aggressive
player who would semi-bluff with something like Q-T (81%) or Qd-9d
(82%).
Having calculated the probabilities of winning, our hero is now
left with the subjective part of the answer, guessing the probabilities
of what the other player has. I would guess that it's more than
twice as likely that the player has two pair, or A-K, or even some
weaker hand than that he has three of a kind. And I would guess
that maybe 5% to 10% of the time, Ad-Td is actually ahead. I told
our hero that, based on the numbers, I would have called.
Our hero did call, and the other player had K-J, giving our hero
a 44% chance of winning the hand. The turn card was the 2d, but
the river was a jack and our hero's flush lost to a full house.
The river card was a tough break, but playing by the numbers, he
still made the right play.
It's good to know the numbers, but it's equally important to know
how to get them. And if you use the available tools whenever you
aren't sure, you'll start to remember them when they come up at
the table. In poker, every tool in your toolbox brings you one step
closer to mastery of the game.
Andy Bloch
Play
Online Poker
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