Probability question

[MathBabe]

How are you planning to force the flops? If you deal the flop first, I don't think that's a reasonable simulation. The best way to simulate would be to deal some huge number of hands, and only choose the ones where AAA came on the flop.

I think in the end we've arrived at the same answer. I said 40% when I used the conditional probability formula. Your mistake, which I have to admit I didn't catch, when fixed comes up with 40% as well.

That is correct. With more then one card, though, you can't use the easy 4/49*20 calculation - that's where you have to get into the combinations.

[Talking Poker]

Q: If I deal a gazillion hands and throw away all the ones that don't have exactly an AAA flop (keeping only those 100,000), what is the difference between doing that and dealing 100,000 flops with AAA and then distributing the rest of the hole cards?

A: Absolutely nothing! The only difference is the insane amount of time it would take to deal all those hands only to throw 99.9%+ of them away.

That logic right there alone should prove that there is absolutely no merit to your "it matters what happens first" line of thinking. If all the non-AAA (or whatever we are testing) cases are thrown away, why even bother doing them? The subset of hands we will be left with will be exactly the same as if we had put the AAA aside, dealt the hole cards, and then flopped the AAA.

And I'm still confused... did you not also say 74%? Where did that come from then?

And more importantly, how about the "real life scenario" that you have yet to address, where I say we have KK and the flop comes _ _ _ (multiple scenarios) and I want to know how likely it is that someone has a pair of Aces for each of them? Do you agree with me that the AA9 flop is safer for us than the A92 flop, for example?

[GTDawg]

In one scenario, you are distributing the hole cards...and then waiting for a AAA flop. You have four aces when dealing the pockets. The second scenario you already used three aces and you only have one ace when dealing the pockets.

No?

[Talking Poker]

Yes. But if we are throwing away all of the hands that don't have AAA flops, what's the difference, other than the amount of time it will take to complete the simulation?

In other words, we have two choices:

Choice 1:
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? YES!
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
Deal 20 hole cards, deal flop, check to see if it's AAA. Is it? No.
etc, etc, etc for a long, long time to get 100,000 AAA flops.

Choice 2:
Just do this 100,000 times:
Deal AAA flop (set cards aside, whatever). Deal 20 hole cards.

In the end, you'll have the exact same 100,000 hands of AAA flops either way.

[Talking Poker]

It doesn't take all that long to run the simulations, but collecting the data afterwards and putting it all together takes quite a while. This is how much I've done so far, and it exactly supports (proves) my hypothesis.

For 100,000 simulated hands...

AAA flop:
#AA dealt - 0 (0%)
#Ax dealt - 40,674 (40.64%)

AA9 flop:
#AA dealt - 840 (.84%)
#Ax dealt - 77,133 (77.13%)

Please note how with less Aces on the flop, there are now more Aces available to be in people's hole cards (2 instead of 1), and therefore the likelihood of at least one of our opponents holding at least one Ace increases. Exactly like common sense tells us.

Please let me know if you'd like me to continue with A92 and so on... I will, but surely this is enough FINALLY to prove my point, yes?