Poker probability question (Texas Hold'em)

[DingDingDao]

So I play in two regular weekly games, and recently I started wondering about heads-up play. Specifically, what are the odds that my opponent does NOT catch any cards on the flop (meaning no pairing cards)? I'm not concerned about draws--I just want to know how often (if I bet into the flop) my opponent will have a pair.

Here's the math I worked out:

P(no pair) = (50/52)^3

The way I reason this is that if my opponent does not have a pocket pair, the odds that he/she catches a pair is 2 in 52. Times three cards on the flop = ^3.

That gives me a probability of P=.888996, or about an 89% probability that I'm betting into an unmade hand. Does this sound right?

 

[chuckywang]

It's more complicated than that. You have to condition on event if you and your opponent pair up. e.g. if you and your opponent both have 6's, than the probability that he doesn't get a 6 on the flop is greater than if you and your opponent don't pair up. There are few cases to consider.

 

[DingDingDao]

Originally posted by: chuckywang
It's more complicated than that. You have to condition on event if you and your opponent pair up. e.g. if you and your opponent both have 6's, than the probability that he doesn't get a 6 on the flop is greater than if you and your opponent don't pair up. There are few cases to consider.

Yeah, I realize that there are other possibilities, some of which are:

1) I have one of his cards (or both)
2) I have a pocket pair that matches one of his cards (or both)
3) He flops a straight
4) He flops a flush

I'm kind of just looking for a quick and dirty solution (ballpark, not an exact probability, which would depend on more factors, like my hole cards)

 

[mchammer187]

if he has 2 cards than he has 6 cards that will make a pair so 44/50

so your 2 in 52 is wrong

and where is the 52 coming because looking from your perspective or the opponennt's 2 out of 52 cards will already be known so you have to consider that as well

I can tell you that it is definitely lower than 89% you are betting into no pair

 

[chuckywang]

Then I guess what you're doing is quick and dirty, except of (50/52)^3, it's 44/50*43/49*42/48. Just factor in the probability that his hole cards are paired, which is 3/51.

So it's something like 48/51*44/50*43/49*42/48 which is about 63.6%.

 

[DingDingDao]

Originally posted by: mchammer187

if he has 2 cards than he has 6 cards that will make a pair so 44/50

so your 2 in 52 is wrong

and where is the 52 coming because looking from your perspective or the opponennt's 2 out of 52 cards will already be known so you have to consider that as well

I can tell you that it is definitely lower than 89% you are betting into no pair

Oh, right *slaps forehead*

Ok, so (44/50)^3 = .6814, about 68%. That sounds more correct.

I guess that probability changes if I pair a card on the flop. That would reduce the probability to (44/50)^2 * (45/50) or worse (if I flop trips or two pair). Of course, if I paired a board card I'd probably bet into it anyways.

 

[DBL]

Originally posted by: DingDingDao

Originally posted by: mchammer187

if he has 2 cards than he has 6 cards that will make a pair so 44/50

so your 2 in 52 is wrong

and where is the 52 coming because looking from your perspective or the opponennt's 2 out of 52 cards will already be known so you have to consider that as well

I can tell you that it is definitely lower than 89% you are betting into no pair

Oh, right *slaps forehead*

Ok, so (44/50)^3 = .6814, about 68%. That sounds more correct.

I guess that probability changes if I pair a card on the flop. That would reduce the probability to (44/50)^2 * (45/50) or worse (if I flop trips or two pair). Of course, if I paired a board card I'd probably bet into it anyways.

You hit a pair on the flop ~1/3 of the time which I guess is supported by the Math above. However, if the board pairs, it is less likely that your opponent has hit the flop, although more likely that he now hold 3 of a kind.

 

[amoeba]

the correct math is (44/50)(43/49)(42/48).