[AaronBrown]

[ QUOTE ]
If we deal a ten-handed round, each individual's odds of getting at least one Ace is around 15%. On the whole, there's about an 87% chance that at least one Ace is out there, but does that mean that if everyone folds to the button, that the button is any more likely than 15% to have an Ace? I'd say no: his cards were independent of the others dealt, and while no one having an Ace is quite unlikely, no one before the button having an Ace is unlikely as well, so it's not like the odds need to "compensate". In some sense this resembles the reasoning of those who think they're "due" after a poor run of cards, though your argument is certainly better reasoned.

[/ QUOTE ]
Thanks for the kind words.

I agree the argument sounds like players thinking they're due, but the difference is one player's hand is not independent of another's on the same deal; the way one deal in independent of the last deal. If none of the first nine players has an Ace, the probability that the last player does goes up from 1 - (48/52)*(47/51) = 15% to 1 - (30/34)*(29/33) = 22%. With fewer non-Aces available, the chance of getting an Ace goes up.

To extend that to my example, assume that players call 80% of the time with an Ace and 20% of the time without an Ace. I know that's math, not Poker, but bear with me. In that case, a player would call 29% of the time, 12% of the time because they had an Ace (15%*80%) and 17% of the time without one (85%*20%). So if the first player calls, you figure there's 12%/29% = 41% chance he has an Ace.

But suppose it comes all the way around to you in the big blind, and only one person has called. You don't have an Ace. This should happen 15% of the time. One possibility is no one has an Ace (13% chance) and one person out of 9 called without one (30% chance) for an overall 4% chance. But it's more likely that one person had an Ace (37% chance) and called with all the non-Ace hands folding (13% chance) for an overall 5% chance. There's 1% chance a person has a pair of Aces and called, 3% chance one Ace was out but only a non-Ace hand called and 1% chance that more than one Ace is out and one of the Ace hands called.

The odds that you're facing an Ace have gone up from 41% to 55%, and the odds that there are at least three Aces left in the deck have gone up from 50% to 78%.

Obviously in real Poker, there are a lot more factors. Some players play loose, others tight. Some hands will raise. Some will call late position but not early. And so on. Still, I think there's value in considering the hands that have folded.

To me, reading is guessing what your opponent has based on his play. Probability is using inferences from the folded hands and your opponents' to guess what's left in the deck.