casino poker chips

Law of Averages in Poker

The "law of averages" is a limit theorem that says the sample mean gets arbitrarily close to the actual mean as the sample gets larger. "Mean" is a technical term for average. "Actual mean" for our purposes is the theoretical average that's implied by a probability distribution; "sample mean" is about actual observed results. A sample mean is the mathematical average of some actual result.

For example, in our black-card red-card probability distribution, think about a black card as representing the value 1 (you win a dollar if the card turned up is black) and a red card is the value 0 (zero). The mean of this distribution is 0.5. On the average you win $.50 for every turn of the card. Half the time the card is black and you get $1, and half the time the card is red and you get nothing.

If we shuffled the deck and turned over a card ten times, on the average we'd get a black card five times, but we might get six black cards, or four black cards, and in some unusual results, we might even get a black card all ten times. Let's suppose that happens, we shuffle and turn a card ten times, turning over a black card every time. What does the law of average tell us about the results from the next ten times we turn a card? The next 100 times? The next 1,000 times?

It actually doesn't tell us anything about the individual results from future events. It tells us something about the overall average for a large number of replications. All the law of averages really tells us is that the sample average won't be affected by any short-run results if we make our sample size large enough.

The law of averages tells us our results should look like this if we start off with ten black cards:

Sample Size Black Cards Sample Mean 10 10 1 100 55 0.55 1,000 505 0.505

Our sample mean gets closer and closer to the actual average of 0.5 because of the increased sample size. The more cards we turn, the less effect on the total those first ten cards have. Luck does not even out. the first ten black cards don't get offset by a run of ten red cards. Luck averages out because in the long run short-term results don't have a big effect. This difference between luck averaging out and good luck being offset by bad luck is important. If you get lucky, then you just got lucky. The gods of chance aren't going to take it away from you.

The law of averages is often misinterpreted to mean such things as "red is due" at roulette because it hasn't hit in a while. That's because people tend to think that the process that causes the sample mean to get close to the true mean is that sequences on one side of the mean get offset by sequences on the other side, that is, that a run of ten blacks will, at some time, be offset by a run of ten reds. However, that's not what makes the sample mean converge to the actual mean. It's just a large sample size that does that.

Remember this the next time you miss twenty flush draws in a row. That is not going to mean that you are due to make the next draw. The cards don't have a memory.