[Tommy]

Andy Bloch Presents "No-limit by the Numbers"

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.