Poker Digest, Vol. 5, No. 4, February 8 - 21, 2002
Last night, the following hand developed in a $5-$10 Texas Hold'Em game in which I was participating. I was in the big blind and peeked at my cards when the action returned to me. Four players had limped and my hand of 9-3 suited made me grateful no one had raised. At least I would get to see the flop.
The rainbow flop of 9-9-A made me even happier no one had raised before the flop. I bet and had three callers. The turn card was a 3, giving me a full house and putting two suited cards on board. Two of the other three players called my bets all the way to the end and I won a nice pot off a weak hand in the big blind.
Later, someone asked me how confident I felt after the turn card had produced the full house. Of course, the answer was ``very confident,'' but let's take a more precise look at the chances of someone beating the full house.
We are going to examine two scenarios. The first is an assumption of facing nine random hands. So, suppose we have a player holding 9-3 with the four board cards A-9-9-3. We need not worry about suits because flushes are not relevant. The only way the player holding 9-3 can lose is if someone ends up with a bigger full house. We also may ignore quads because the only possible quads are four aces. If someone has pocket aces, they already are ahead anyway.
First, we determine the probability one of the other nine random hands is ahead after the turn and before the river card. This is the case only if a player holds either A-A or A-9. There are ways of choosing nine random hands. Counting the number of ways at least one of A-A or A-9 may have been dealt, we obtain a probability of approximately .051 that at least one random hand is ahead after the turn card.
If another player is holding A-9, then the player holding 9-3 cannot catch up. However, if a player is holding A-A and no one has been dealt a 9, then the player holding 9-3 wins if the case 9 comes on the river. While unlikely, there is a probability of approximately .00059 that this occurs.
There are more likely ways for the player holding 9-3 to lose on the river: One of the random hands may consist of a pocket pair of rank 10, J, Q or K and the river card matches the rank of the pair, or a player holds 9-x, with x being bigger than 3, and a card of rank x comes on the river, or a player holds an ace and another ace comes on the river.
The probability of the river card having rank 10, J, Q or K and one of the random hands being a pair of that rank is about .009. The probability of one random hand holding 9-x wth an x coming on the river is about .023. The probability of someone holding an ace with an ace coming on the river is about .042.
The probabilities calculated above are not independent so that taking their sum creates a more pessimistic situation than the truth. The sum is .124, leading me to guess the true answer is somewhere around .1 or rough odds against losing of about 9-to-1.
Let's now consider the actual play of the hand under discussion. Given the players in this particular game, had anyone held a pocket pair of rank ten or bigger, there would have been a preflop raise. Thus, we can be almost certain that there were no large pocket pairs in any of the four players' hands. Thus, that particular avenue of escape for the other players was closed.
Because of bad-beat jackpot considerations, most players at the table would have played A-9. But I was first to act after the turn card, and since no one raised my bet, I was certain no one held a 9, let alone A-9. I assumed I was facing one or two players with some kind of ace. I don't believe all three callers held aces. The latter is unlikely and given the tendency for several of the players to chase flushes -- even if they catch only one card of their suit on the flop -- it likely was the case that one of the players was looking for a flush.
Under the assumption I was facing two players with an ace, the probability was only 1/28 that the case ace would appear on the river. Under the assumption I was facing only one player with an ace, the probability was 1/14 that an ace would come on the river. In both situations, my hand was a big favorite to hold up.
On those rare occasions when a poor big blind hand turns into a full house, the preceding numbers give some kind of indication just how strong your hand is.