Texas Hold \'Em to Solve This Problem,

Texas Hold 'em

To solve this problem, there are a couple of steps. These steps are based on the principle of expected outcomes, which are played out over the long run. So the first step is to calculate the odds of receiving a card on the river that could win me the hand. The second step is to calculate the expected payoff based on those odds. If this payoff is better than the $10,000 I would have to call at this point. Past money in the pot is not relevant because it has already been bet.

This strategy will work because it is based on mathematical principles that hold up over the long run. In order to win at poker in the long run, short-term results -- a loss here for example -- must be ignored. If a bet is greater than the expected payoff if I win, that means that in the long run I will lose money. If I make a bet that has a higher expected payoff that the size of the bet, then in the long run I will make money as a poker player.

My opponent has three of a kind with 10s. I can win with a straight, but I could also win with a flush. Because suits are not given in this example, we will assume that there is no possibility of a flush. If there was a chance at a flush, that would obviously change the odds and the math and it would probably increase my odds of going all-in.

I know 8 cards, which means that there are 44 cards left in the deck. To get a straight, I need an 8 to come up on the river. There are four 8s that are unaccounted for at this point. I need a straight to win, because with his pocket 10s and the 10 on the turn, he has the better hand right now. I need the straight in order to win. Another ten would improve his hand, but he will beat me anyway unless I get an 8 for the straight. Anything other than an 8 would not help me, because I cannot match a three of a kind with anything else in my hand. Knowing what he has helps me immensely, because it takes strategy out of the question and makes it a simple math problem.

3. So the odds of an 8 coming down the river are 4/44 = 9.09%

The pot, with his $10,000 raise and my $10,000 call, will be:

($24,000)(2) + ($10,000)(2) = $68,000

The expected payoff from going all-in here is (.0909)(68,000) = $6,181.81.

4. At this point, I should fold. My odds of winning are long, and what will happen is that the expected payoff from this bet -- given those odds -- is less than the bet. Thus, folding is the rational option here. Prior to the turn where he got his three of a kind, I was still in a bad position because he held the highest pair -- I would have needed two pairs or a three of a kind just to beat that, or the straight of course.