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# Roulette: After 3 Reds, Do You Pick Red Or Black?

April 22, 2011 10:32 am

[This post is based on a great paper from Voxeu, titled ‘Predicting the Improbable’.]

All of us have a tendency to over-react to recent events. It is obvious in many of our actions, starting with what we talk about all the way to where we spend or invest. But the reaction could be in two ways – we see the trend, and either think that the trend will break, or that the trend will continue. Why is this important? Turns out, this is the easiest way to classify people: take them to a Roulette table that has thrown three consecutive Reds. Do they bet on black or red?

“There are two plausible but apparently contradicting intuitions about how people over-infer from observing recent events.

The “gambler’s fallacy” claims that people expect rapid reversion to the mean.

For example, upon observing three outcomes of “red” in roulette, gamblers tend to think that “black” is now due and tend to bet more on “black” (Croson and Sundali 2005).

The “hot hand fallacy” claims that upon observing an unusual streak of events, people tend to predict that the streak will continue.”

But the biggest learning for us is this: the number of times Red turns up or Black turns up will become approximately equal only when the number of turns on the Roulette wheel is very high, running into Millions of turns..

For any of us to expect that over a small number of turns this probability will work is just being stupid. The paper puts this succinctly:

“A person believing in the “law of small numbers” thinks that small samples should “look like” the parent distribution, i.e. that the sample should be representative of the parent distribution. Thus, the person believes that out of, say, 6 spins 3 should be red and 3 should be black (ignoring green). If observed outcomes in the small sample differ from the 50:50 ratio, immediate reversal is expected. Thus, somebody observing 2 times red in 6 consecutive spins believes that black is “due” on the 3rd spin to restore the 50:50 ratio.”

Think about it – we all believe in the law of small numbers one way or the other. In all fields. We are just being stupid.