Probability of a flush draw being out

[asdf1234]

I was just thinking about how likely it is that a flush draw is out given a board with two of a given suit on it. I made the calculation, but need confirmation on it.

Say I hold Ac Ad, and the flop is
Kh 6c Qh

I want to figure out how likely it is that somebody holds two hearts.

Okay, so there are 11 hearts left, and that makes 55 possible combinations, correct?

Now, there are 47 unknown cards for a total of 1081 possible hands.

Would it just be 55/1081 = .05087?

5% chance?

I think this only works for playing against a single hand, right?

So, to figure out the likelihood that any opponent out of say 4, holds a flush draw, just calculate the probability that none of them holds one?

(.949)*(.949)*(.949)*(.949) = .811

Any help appreciated. Thanks.

[BruceZ]

So, to figure out the likelihood that any opponent out of say 4, holds a flush draw, just calculate the probability that none of them holds one?

(.949)*(.949)*(.949)*(.949) = .811

That is a close enough approximation. It is not exact because the hands are not quite independent, so you cannot just multiply these probabilities for an exact answer. Another approximation would be to just take 4 times the probability that one player has them, 4*55/1081 = .2035, so the probability that no one has them would be .7965. This is also approximate because two or more players could have them, and this would overcount. One of these approximations is too small, and the other is too large, so you know the exact answer lies in between. Since they are close, you know we aren't off my much. To get the exact answer, you need to use the inclusion-exclusion principle, described below and elsewhere. I computed this, and it took about 1 minute, but I'll let other's elaborate and in the process find out if anyone else has learned how to do this from my posts. This is an especially good one to practice on since the computations are very easy. The correct answer, to 9 significant digits, came out to 80.7413988% that no one has them, and 19.2586012% that someone has them. I only give this many decimal places so people can check that they did it exactly right. I don't normally believe in being this ridiculously precise and quoting God-awful long strings of meaningless digits as certain people are wont to do, especially in cases when the least significant digits aren't even correct due to finite precision effects (they are correct here). Normally 1 or two decimal places, hey, it's all you need.

For a more meaningful result, you would normally not assume that the other players are playing random hands.

[Cyrus]

Some things will never change, I see. /forums/images/icons/cool.gif

"Normally 1 or two decimal places, hey, it's all you need."

In real-life situations, most of the time we don't even need one decimal !

In our arithmetical calculations, we usually need much more than one decimal, if only to be able to re-check if the calculation "is exactly right".

A trivial example, perhaps unknown to certain people, is when computer programmers are comparing results from their home-built Blackjack combinatorial calculators. Don Schlessinger, who is rightfully held in high esteem by certain people here, had questioned at one time the usefulness of quoting a myriad of decimal digits, only to be informed that they are absolutely necessary to make sure that the different programs come up with the same answer.

"I don't normally believe in being this ridiculously precise and quoting God-awful long strings of meaningless digits as certain people are wont to do, especially in cases when the least significant digits aren't even correct due to finite precision effects."

Well, I am not in such close terms with God as the recent movie hit will have certain people to be, but if anyone was curious about them "long strings" of digits in my posts, one could have just asked : Turns out it's simply a copy & paste job from the calculator's screen. /forums/images/icons/grin.gif

--Cyrus

PS : I wonder why that calculator is supposed to suffer from "finite precision effects", though...

[Cyrus]

The name of the well-known Blackjack author and all-round authority is spelled, of course, Don Schlesinger. Don will not forgive me for spelling it Schlessinger but there are certain people here I dare not contradict.

/forums/images/icons/grin.gif