At the roulette table, black comes up six times in a row. The gambler thinks, “Since half the time black comes up and half the time red comes up, the wheel is due to land on red,” and bets on red. This is the gambler’s fallacy.
Why is the gambler wrong? Isn’t it true that half the time the wheel will come up black and half the time it will come up red? These are some loaded questions. It is true that given an infinite number of spins of the roulette wheel, red and black will each come up half the time. Even in only a few spins we expect the wheel to come up red and black relatively equally. However, this certainly doesn’t happen every time. At any given point in time, it just as likely to spin a red as it is to spin a black, regardless of how the previous turns came up. That is, the probability function is memoryless.
So why is it that people expect the wheel to ‘even itself out’?
We suffer from a clear misconception that the spins of the roulette wheel should look random. We rely on representativeness heuristic to determine what is random. For example, if we flip a coin ten times, here are two of the possible sets of outcomes:
(Trial A) T H T T H T H T H H
(Trial B) T T H H H H T H H H
Trial A looks more like what we would expect at the results to look like because the Heads and Tails lookrandom. In actuality, each specific result is equally likely (as is any specific string of Heads and Tails). Since we expect the results to look random and to demonstrate the equal probability of flipping Heads or Tails, many people will assume that Trial A is more likely. They also assume that Trial B is due for a Tails (note only three of the ten flips landed Tails while seven landed Heads).
We can explore this in a graphical sense. Imagine two squares each with ten dots. Each is a random distribution of the dots throughout the square, however one square's dots look more random than those in the other square because the dots are more spread out. However again, each of these two distributions is equally likely. In fact, there should be no expectation that the pattern even appear spread out. This is the beauty of random selection and random distribution. Anything and everything is equally possible (that is, any specific event). Thus, a gambler should not wage his chances based on the history of the roulette wheel. Each turn is a new chance, a new trial, a random spin.
