it lets me explain poker tournament odds based on a strange mathematical truth. But first I’d like to talk about the whole notion of mathematics in poker.Too many players worry that they aren’t good at calculating poker odds. They believe that this failure will greatly diminish their profits or cause them not to win at all. The truth is that you don’t need to calculate poker odds.I’ve never known a world-class player who did complex calculations during a poker hand. I don’t.Why? It’s because most poker strategy is obvious to me. It’s been pre-calculated away from the table. About the only thing I do with math at the poker table is something like this: I reason that the pot is $10,000, it costs me $3,000 to call and see the river card, and if I do, there are nine cards left that will make my flush, out of 46 unknown cards.So, that’s 37 cards that will miss for me against 9 that will hit. The 37-9 reduces to a bit worse than 4-to-1 against me.I’m only getting 3.33-to-1 on the money, and my odds against making the flush are longer than that. So, should I call?Here’s where I take other factors into consideration to see if I can make up the statistical shortfall. How much extra, on average, can I earn on the final-round wagering if I make the winning flush? How much chance is there that I could successfully bluff on the river? What is the chance that I might make a winning pair? What chance is there that I can make my flush and still lose — perhaps to a bigger flush or to a full house or four-of-a-kind if the board pairs with a card of my suit?The point is, I’m only going to be able to guess; the mathematics will deteriorate to an estimate. And that, despite what you’ve been told, is how most poker decisions are made — educated guesswork. So, if you’ve been wondering how non-mathematical players sometimes can fare well against statistical wizards, that’s why.Reasonable choices are obvious for most situations. It’s the quality of your estimates that determines how often you do the right thing.Although I seldom play poker tournaments, I’m here at the World Series of Poker to do a private seminar for Doyle’s Room players. Since I’m here, I entered a few events and, by golly, I stumbled upon a new policy. They’re letting players enter up to two hours after an event starts.You get to take a seat in the event that’s already underway and start with your full amount of chips. Advantage or a disadvantage?On the disadvantage side, you can point out that by sitting out the first couple hours, you’re missing a chance to gather chips. This is especially meaningful if you have an edge against the average field of players. And since weaker players tend to get eliminated early, you’re joining a slightly stronger field, on average, when you enter late. Even if only a few players have been eliminated, the stacks usually have shifted toward the stronger players, so not as many chips are in the hands of weak foes.But those considerations may be overshadowed by a simple mathematical truth. In a proportional payout tournament — where first-place only wins a fraction of the prize pool, second-place a smaller fraction, and so forth — your chips are worth more and more as the tournament progresses.In that light, let’s revisit the hand we talked about earlier. Although it might be profitable to play that flush draw in a regular non-tournament game, it isn’t profitable to play it in a proportional-payout tournament.Now let’s return to our discussion of whether to exercise the option to buy in late. A sophisticated player once told me he wouldn’t waste his money rebuying late in a tournament for the original entry price.He reasoned that the average stacks opposing him would be almost insurmountable after so many other players had been eliminated and donated their chips to the few remaining players.That may seem logical at first, but it isn’t. Let’s say it’s a standard proportional payoff tournament in which first place gets $1 million, second place gets $500,000, and third place gets $300,000. If you could buy-in for $10,000 when there were only two players left, would you? Of course! You’d take third place automatically — and maybe better.Of course that’s a silly example, based on something that could never happen. You can buy in up to two hours late, but not at the final table.Fine. The point is, the later you buy in, the more your chips are worth. So, other things being equal, it’s a slight mathematical advantage to enter those big events late.http://pokerstars.api.pstv.videojuicer.com/presentations/9735.html