Basic probability question...

[kurto]

I apologize if you've seen this a million times, but I last took a math course over 15 years ago and I've forgotten a lot...

How do you compute odds with 2 cards to come?

I know if I have 10 outs on the flop... I'm 10/47 to make it on the river, and if I miss, I'm 10/46 to make it on the river.

But what's the equation to figure out what percentage of the time you're going to hit when you combine both cards to come?

How does one add multiple statistical probabilities to get one number? (for that matter... I'd be curious how to do it with more then 2 cards. Like If I flip a coin 10 times... what are the odds you never once land heads?)

Thanks.

[Cobra]

The easiest way to do this is with combinations. The probability of hitting one or more of your outs is the same as 1- the probability of hitting none of your outs.

10 outs
47 unseen cards
37 blanks

Using combinations it is 1-(37 choose 2)/(47 choose 2) which is equal to 38.4%. That is one minus the total two card combinations from your 37 blanks divided by the total two card cominations of the unseen cards.

It can also be done 1-37/47*36/46=38.4% or 1 minus you get one of your blanks on the turn and one of your blanks on the river.

There are other ways of doing this problems but I believe these two are the easiest.

Cobra

[Crooked Paul]

Well, to answer your original question:

"How does one add multiple statistical probabilities to get one number?" (emphasis added)

Guess what? You multiply. If you have a fair coin, the chance it will come up tails once is 1/2. The chance that it will come up tails twice running is 1/2 times 1/2, or 1/4. The chance that if you flip it 10 times it will come up tails all ten times is:
1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2
= (1/2)^10
= 1/1024

If you really want to get into it and figure out your chances exactly (which I do recommend), I'd advise you to think out the problem for a second and write out the different success conditions, paying careful attention to compound events (which is when one chance depends on another, and you have to multiply -- like all the coin-flipping above).

So, like you say, if you have 10 outs on the flop, you want to be adding the chance you'll hit one on the turn (10/47) to the chance that you don't hit on the turn AND hit on the river, which is a compound event:
turn chance + (turn miss * river chance)
(simple) + (compound)
10/47 + (37/47)*(10/46)

I'm not going to work out all the math. I think you see my point.

But dude, check this out! Who wants to do all that math in the middle of a hand? There's a shortcut. Count up your outs, then multiply by 2 to get your rough percentage to hit on the next card (works on either flop or turn), or multiply outs by 4 to get the rough percentage to hit with two cards to come (use this only if you're going all-in or you know you intend to call any bet on the turn).

So in your example, if you have 10 outs on the flop, you've got about 20% to hit on the turn or 40% to hit on the turn OR river. This shortcut is invaluable.

The reason it works (maybe this will help you remember) is that at each stage of your decision-making in hold'em, there are roughly 50 unseen cards, so each one of your outs represents roughly 1/50 chance, or about 2%.

Hope this helps.


Crooked