How Probabilities Change with Each New Card

If a player raises from early position, he's 61% to have an ace (we'll motivate this number below.) If an ace then comes on the flop, suddenly that same player is only 53% to have hit it!

This may seem completely nuts, but it's the correct way to reason in poker. I'll show you why.

Example: early position raiser

Let's say that this guy always raises from early position if he has AA-TT, AK and AQ. He never raises with anything else and he never limps from early position.

Since AK and AQ are more common than the pairs (they come in more combinations) he'll have an ace in his hand 38 times out of 62, that is, 61% of the time. See the numbers in the table:

Pre flop raiser

An ace on the flop

Now, if the flop comes down with one ace, the number of ways in which the opponent can have an ace in his hand diminishes.

With one ace gone, there are only three combinations of AA left, and the number of AK and AQ go down to 12 each.

Ace hands are now 27 out of 51, that is, 53%:

With an ace on the flop

Two aces on the flop

What if there are two aces on the flop? Then there's only one AA combination left - the two case aces - while AK and AQ come in 8 variations each. The probability for the early position raiser to have an ace is now 17 of 41, or 41%:

Two aces on the flop

Now you may say that 10% more or less isn't that big a deal. Maybe you're right, but I'd say that the underlying principle is important. It's active all the time, changing everything as soon as a card hits the board.

If you're in the hand with KK against this early position raiser, the chances of him holding an ace are pretty important for your decisions. Forty or fifty percent can make all the difference in the world, depending on pot odds and other factors.

The opponent's behavior comes into the picture, of course. If two aces come on the board and he seems weak - or strong - this observation adds to the equation.

The poker pros do it - WSOP example

Professional players use this thinking a lot. When the chances of the opponent holding a better hand goes down enough, they're always prepared to gamble it up.

One player reported the following story from the 2008 WSOP Main Event. He limped in early position with J5 and a player in middle position raised it up. Our guy called, thinking that he would be able to outplay the opponent on later streets. He's a pretty confident guy.

When the flop came AJ8, he figured that the chances of the opponent holding an ace went down enough that he could move in on him with good expectation.

He reasoned that with AJ on the table, the opponent was more likely to have KK-QQ-TT than any hand containing an ace or a jack (a jack least of all since he had one in his hand too.) And with the ace on the board the opponent would be forced to fold KK or QQ.

In this case, the opponent had JJ and our guy was eliminated on Day 1. But, as you should know by now, a single outcome doesn't disprove a probability distribution.

Maybe he took this thinking a step too far, or even misused it. Or maybe he was right, and he'd make a win most of the time. He's a winning payer, after all.

There's no final answers in poker, but the principle described here is hard core, and if you you use it wisely, it will make you a stronger player.

/Charlie River

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