On August 18th, 1913 something unusual occurred at a roulette table at the Monte Carlo Casino. At a particular table the ball had fallen on black many times in a row. After each successive black result gamblers placed more and more money on red thinking that after so many blacks in a row that red was “due.” The ball finally fell on red on the 27th spin. Millions of dollars had been lost by that point by gamblers betting on red who had suffered from a common mathematical misunderstanding.
The chance of a European roulette wheel coming up black 26 times in a row is 1 in 145 million. Thus, it would appear that each spin of another black was increasingly less likely as the chances of 27 blacks in a row is 1 in 298 million. Of course, this is not the case – each spin is independent of every other spin and the chance of a black vs. a red is nearly 50/50 (a European wheel has 37 slots, 18 red, 18 black and zero). Thus, the chance of a Red on the first spin is 1 in 2.06, the same as it is on the 10th spin and the 27th spin.
Here’s the common misunderstanding- if we know that a series of random/independent occurrences should have an expected average value, we consciously or unconsciously expect that future occurrences will act to “force” the series of observations back to their expected average.
For example, assume a fair coin is flipped 20 times with 16 heads and 4 tails. What is the expectation for the next 20 flips? Should we expect more tails to pull the average number of heads/tails closer to 50/50? That may be our gut instinct, but that is wrong. The coin has no memory of the prior 20 flips and there is no force that will cause a greater number of tails.
Instead, the law of large numbers says that as number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero. In other words, as you have more and more observations of a fair coin, the actual experience of heads and tails will approach your expectation of 50/50. This is a subtle difference – there is no “force” causing more tails after a string of heads – it’s just random variability.
Imagine flipping a coin one million times. Actual observations of heads/tails will be very close to 50/50 because one million observations is a large number. However, it is not unusual to think that within that million coin flips there could be a string of 10 heads in a row, or 16 heads out of 20 flips, etc. All sort of random patterns will occur within that string of one million flips. Our problem is that our experience with coin flips or roulette spins or most things in life is that we deal with very small sample sizes which may contain great variability.