Wheel of fortune
Wheel of fortune
Definition
Marge was no mathematician, but she knew she had just discovered a foolproof system to get rich playing roulette. She had been observing the spin of the wheel at the casino for several days. During this time she had noticed that it was surprisingly normal for there to be a sequence of spins when the ball fell into only black or only red slots. But five in a row of the same colour was very unusual and six in a row happened only a couple of times a day. This was going to be her system. The chances of the ball falling into a slot of the same colour six times in a row were tiny. So, she would watch, and once it fell into, say, red, five times in a row, she would bet that the next one would be black. She was bound to win more often than she lost because six in a row was so rare. She was so confident that she had already started to think about how she would spend the money.
Source
TBC
Motivation & Background
Marge’s mistake is a warning against the limits of thought experiments. If her system seems foolproof, it is because she has already tested it out, and it works every time. In her head, that is. If the gambler can be so easily led astray by imagining what would happen in hypothetical situations, so can a philosopher. Her mistake, however, is one of reasoning, and is not caused by any failure of the real world to match the one of intellect. The mistake she makes is to confuse the probability of the ball falling into the same-colour slot six times in a row with the probability of it falling into the same-colour slot, given that it has already done so five times in a row. Imagine, for example, a simple game of luck where people compete with each other on the toss of a coin. In round one there are sixty-four people, round two thirty-two, round three sixteen and so on until in the final there are just two. At the start of the contest, the chances of any given person winning are 64–1. But by the time you get to the final, each remaining contestant has a 50–50 chance of winning. On Marge’s logic, however, the odds are fixed at round one. And so, in the final, although there are only two people left, Marge would reason that each one has only a 1 in 64 chance of winning. Which would mean, of course, that that there is only a 1 in 32 chance of either person winning! To return to the roulette wheel, it is indeed very unlikely that the ball will fall into the same colour slot six times in a row, just as it is very unlikely (64–1) that any given person will win the coin-tossing contest. But once the ball has fallen into the same colour slot five times, the initial improbability of a sequence of six is irrelevant: for the next spin of the wheel the chances of the ball falling into either red or black is a little less than 50–50 (there are also two green slots on the wheel). The point is that the improbability of what has happened in the past does not affect the probability of what is yet to happen. Marge should have seen this. Had she observed how frequently a series of five of the same colour extended into a series of six, she would have seen that the chances were in fact, a little less than 50-50. Her mistake, then, was not just one of faulty reasoning, but of imagining something to be the case that her observations could have confirmed was not. She is a poor experimenter, in her head and the world.
