Waiting Time

There are certain mathematical mistakes that poker players, and unfortunately poker writers, frequently make. These errors creep into some books and articles which contain a lot of conclusions based on mathematical models. And while most of these mistakes are not really crucial in themselves, they always make me worry than there may be flaws in some of the other mathematical conclusions they presented as gospel. It is common to read that typical waiting time for aces is 221 hands. In actual fact, this is the average wait time. The typical wait time is much less.

One common error involves the wait time for something to happen. The average wait time is frequently confused with the median or mode wait time. Let’s begin with a non-poker example. When you drive over to pick up your girlfriend, you are usually right on time. Once, however, you had a flat and were 150 minutes late. After you have picked her up five times, your average lateness is thirty minutes. (4 times 0 plus 1 times 150 equals 150. 150 divided by 5 equals 30 minutes.) The last five times she has come to pick you up, she has always been exactly 30 minutes late. You both have an average lateness of half an hour, but you are a big favorite to be on time, since your median time and your mode time are both zero. She is not. (The median is found by arranging a set of observations in numerical order and picking the middle one. The mode is just the most frequent observation.) Any gambler would want to bet that you will be on time and she will be late.

So far we have looked at the waiting time for your girlfriend. Now we’ll look at the waiting time for something you really love – aces in hold’em. Let’s begin by calculating the average wait time. There are 1326 possible starting hands. This is calculated as follows: the first card may be any of 52 and the next any of 51. 52 times 51 equals 2652. But for our purposes order doesn’t matter (6 5 is the same as 5 6) so this number is divided in half. 2652 divided by 2 equals 1326. There are 6 pairs of aces. (A A, A A, A A, A A, A A, A A) This means the odds of getting aces are 221 to 1. (1326 divided by 6.) How long will you have to wait to get aces? On average 221 hands. You will generally get them much more frequently than that, but occasionally you will have a long wait. So what is the median wait for aces? It is around 155 hands. To do this calculation you need a calculator or computer that can handle exponents. Basically, the chance of getting aces is .0045 (1 divided by 221.) This means that the chance of not getting aces is .9955. The chance of not getting aces on either of two hands is .9955 times .9955. This is .9910. Note that the number is getting smaller. If we keep multiplying by .9955, it will get smaller and smaller. Eventually it will reach .50. This is the point at which you are even money to have had aces at least once. It turns out to be around 155 hands. That is, the median wait for aces is 155 hands. If we lower our expectations a little, and wait for aces, kings, queens or ace-king, we will be a favorite to have had one of those hands after 27 hands, three orbits.

If you read that you will have to wait 221 hands to pick up aces, you can confidently bet that you will do it much sooner than that. This leads to the following sucker bet. Last year, I dropped by the house of a friend of mine who was playing online poker and cursing his bad luck. I told him that I knew he was much luckier than average, and that I’d bet he got aces more often than most people. He knew that he should get aces once every 221 hands. When I offered to bet one hundred dollars that he’d get aces within 200 hands, he was so eager to bet that he insisted on betting one thousand. Of course I agreed, since the odds were nearly 3-to-2 that he would pick up aces in 200 hands. It took only 100 hands or so for him to pick up aces. (He ending up losing a large pot with them to 7-6 suited.) I graciously paid for lunch out of the thousand I’d won, and not so graciously listened to him rant about how bad his luck was compared to mine. ♠

Steve “Zee” Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. When escaping from poker, he hangs out in his bars on Avenue A — Nice Guy Eddie’s at Houston and Doc Holliday’s at 9th Street — in New York City.