| Why don't we discount dead cards?
New Year's Resolution was to stop playing like a moron, so I spent time over the holidays reading Theory of Poker, Small Stakes Hold 'Em and HOH. But one thing I can't stop wondering is why in the calculation of odds don't we discount the cards that are already dead?
Consistently, when you have four outs on the turn, you're odds are listed as roughly 10.5-to-1. The calculation is 4/46, right? Four cards to save you and forty-six unseen cards.
But because the deck isn't replenished (once a card is dealt, it is no longer possible to come back out), the 46 figure is incorrect. Rather the number needs to be discounted by the possibility that once of those 4 outs has already been dealt -- either to an opponent, or burned before the flop, turn, or river.
I don't have a good probability text handy , but I seem to recall that given a random distribution, the probability that any one of your outs is dead should be something like [out!/(dead-out)!]*out! and this figure should be divided against the "probability" of a card coming on the river which would be the number of outs divided by live cards, multiplied by the probability that any of the outs is dead to the power of the number of outs.
Aside from the exact math -- which I desperately need to check -- there are two implications: the odds commonly tossed around are probably a bit on the high side, meaning that marginally profitable calls are probably actually losing calls (you should be sure the odds are reasonably in your favor if you're behind), and the odds are probably worse the more people there are at your table. Basically, at a nine-handed table, there are eight more dead cards than at a five-handed table (the two cards dealt to your additional opponents).
In light of the posts on schooling, etc., I'm just wondering why this hasn't been factored in to the calculation of pot odds, since on of the aspects of schooling -- not only that your opponents have a greater chance of hitting on their live junk, but also they may have your needed outs -- is valid whether or not they fold or call at any point in the game.  | 
