Practical Probability - Part I

This column is the first of a series on probability. My intention is to give you some basic understanding of probability theory and discuss some of its practical applications. Mathematical thinking in general is unnatural for most people. I am not going to strive for mathematical rigorousness, but will try to help improve your thinking process in some common situations in poker, gambling, and life. As much as possible, formulas and equations will be avoided.



One of the most important attributes of any successful gambler, especially a successful poker player, is his ability to estimate probabilities accurately. He must also, of course, take the proper actions dictated by his estimates. There will be more on this later. First, I want to give you a simple situation in which you are asked to estimate some probabilities. The probability of an event can range from 0 (zero), it can't happen, to 1, it must happen. Given a normal 52-card deck, and someone who plays poker every night for three hours, the probability that he will be dealt the K is 1. Over time, everyone will eventually be dealt every card, and if you play long enough, you will be dealt every one of the 1,326 two-card starting hands. On the other hand, the probability of being dealt two K on the same hand is 0 (zero), it won't happen. Probabilities can be written as fractions, decimals, or percentages. The probability of picking a spade is ¼ or .25 or 25%.



On the first day of a big poker tournament, you are seated next to a young man named Larry. He seems to play quite well, but you haven't really played with him enough to be sure. You end up at the same buffet table during the dinner break and start talking. He tells you that he just turned 21 and that this is the first live casino tournament he has ever played. Now I am going to list a bunch of possibilities about Larry, and I want you to fill in the blank with an estimate of the probability (a number between 0 and 1) that each is true. If you think a statement figures to be true most of the time, estimate .85 or 85%. If this sounds stupid, be patient.

I sincerely hope that you estimated the probability of each of these statements being true. The probability that something happens can never be smaller than the probability that not only does that thing happen, but something else also happens. For example, the probability that you will be dealt an ace is obviously greater than the probability that you will be dealt an ace and a king. Some of the above statements fall into the category of compound events, which can't be as likely as just one of the events occurring. Larry has long hair is more likely than Larry has long hair and a moustache. Larry works at Starbucks is more likely than Larry was an engineering student, dropped out, and now works at Starbucks. Larry's father plays poker is more likely than Larry and his father played poker a lot when he was growing up. Larry plays online poker is more likely than Larry is a winning player online. Larry is a winning player online is more likely than Larry has won more than $20,000 since he started playing online. Check your answers to these statements. Were there some for which you made the error of assigning a higher probability to the compound event than the single event? Amazingly, everyone in the admittedly small sample of my friends who did this exercise made at least one of these errors, and assigned a higher probability to one of the compound events than to the corresponding single event. This includes some people who would never make this kind of mistake if they were presented with a problem involving cards or dice.



There is one other trap into which some people fall when they estimate probabilities for the statements. The probability that Larry has blue eyes plus the probability that he doesn't have blue eyes must equal 1. That seems obvious, and I can only guess that the people who missed it were careless. If you missed it, remember that the probability that something occurs plus the probability that it doesn't is always 1. I am 100% certain that the Giants will win the Super Bowl this year, or they won't.



Now I want to describe a hand I saw played in the Full Tilt Sunday tournament, and discuss how considering that compound events are less likely might have prevented a player from making an error. (All of the amounts are approximations.) With blinds of 20-40 and 5,000 stacks, a middle-position player raised to 120 with the A A. He was called by the big blind (BB). The flop was J 7 2. The BB checked, the aces bet 200, and the BB raised to 500. The aces just called. I don't know if he was afraid of a set or was trapping. The turn brought the A. The BB led out for 800 and the aces, now with top set, made it 2,000. The BB called. The river brought the K. The BB checked, and so did the set of aces. Yes, they checked it down. The BB showed the A 7.



I can't imagine a clearer instance of missing a bet on the river. I know the BB might have backdoored a flush or even a straight, but it was unlikely that he played a hand that strongly that could have done so. Maybe he had the Q J. But for the final check on the river to be right, several events needed to have occurred. First, the BB had to play a hand that had the potential to make a flush or straight in the manner that he did. Second, it had to hit on the river. Third, the BB had to choose checking on the river after he hit the nuts. I would estimate that a river bet of 1,200 would be called nearly all the time, and a move-in also would be called fairly often. Let's say betting with the winner is worth 1,000, and that is being conservative. I would be surprised to ever lose this hand, but let's say that happens 10% of the time. That means that 90% of the time, I win 1,000, and 10% of the time, I lose 2,400. By betting, the player would gain at least an average of 660 (90% of 1,000 minus 10% of 2,400).



I consider this final check to be a conceptual error. The player with the set of aces panicked at the sight of the flush. He didn't understand that the combination of circumstances required for his opponent to have hit the flush and then to have decided to check it made it extremely unlikely that he would lose the hand to a flush.



Steve "Zee" Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. He can be found at many major tournaments and playing on Full Tilt, as one of its pros. When escaping from poker, he hangs out in his bars on Avenue A – Nice Guy Eddie's on Houston and Doc Holliday's on 9th Street – in New York City.