Learning the math, have a question.

[OrianasDaad]

I'm trying to learn the math, but am having some trouble wrapping my head around the math. One question that I have posed myself is:

Holding AK against two opponents, what is the probability that one of my two opponents has made a pair on the flop?

I'm assuming, at first, that between my two opponents they hold four different cards.

If someone could walk through the math, I'd really appreciate it, but even giving the formula would be a big help, as I can probably figure it out from there.

[BruceZ]

[ QUOTE ]
Holding AK against two opponents, what is the probability that one of my two opponents has made a pair on the flop?

I'm assuming, at first, that between my two opponents they hold four different cards.

[/ QUOTE ]

Asssuming 3 different ranks on the flop, with no A or K on flop, take 1 minus the probability that they make no pair:

1 - C(38,4)/C(47,4) = 58.6%.

This assumes they play random hands. Not very useful, but OK for learning.

[Holdem]

ive notied a lot of questions in this probability section that i really dont think have any value. when your up against 2 opponents with AK and your wondering if either of them hit a pair... you'd do much better to try and put a read on them, watch their actions, their demeanor, etc etc etc. simply knowing that they are going to pair up almost 60% of the time really wouldnt do much good to me at all.

[AaronBrown]

There's a lot of truth to this, but remember you also have to make decisions before anyone sees the flop. Simplifying things a bit, assume they'll fold if they don't pair or if an Ace or King hits, otherwise they'll call one bet. That assumption allows you to compute your profit with an Ace, King or no pair for opponents; against which you have to weigh the possible losses playing AK with no help from the flop against a pair.

You have to figure that kind of thing out in advance, so you know when you fold, call, raise and reraise preflop. General rules are too imprecise, you can't tell a bet with a +5% edge from a bet with a -5% edge by watching the game. Over time, that difference can make you a winner instead of a loser.

Once the flop hits, knowing the probability won't help you play the hand. But if you notice a guy acts like he hits 80% of the time, and you know the true probability is 30%, you know something about his play, without the expense of calling him to showdown.