MAKING STATISTICS INTERESTING: The Gambler's fallacy.

The Gambler’s Fallacy

 

Roulette

 

A gambler is betting on what he thinks is a fair roulette wheel. The wheel is divided into 38 segments, of which:

18 segments are black.

18 segments are red.

2 segments are green, and marked with zeroes.

 

Imagine that there has been a long run-a dozen spins-in which the wheel stopped at black. The gambler decides to bet on red, because he thinks:

 

The wheel must come up red soon.

The wheel is fair, so it stops on red as often as it stops on black.

Since it has not stopped on red recently, it must stop there soon. I’ll bet on red.

 

The argument is a risky one. The conclusion is, “The wheel must stop on red in the next few spins.” The argument leads to a risky decision. The gambler decides to bet on red. There you have it, an argument and a decision. Do you agree with the gambler? Since we are talking about the “gamblers fallacy” there must be something wrong with the gambler’s argument. Can you say what?

 

FAIR

 

Arguments have premises. The gambler’s main premise was that the roulette wheel is fair. Fair? What is fair? The word has quite a few meanings.

A judge may be a fair.

A fair wage.

A fair settlement.

A fair grade in this course.

A fair game.

A fair coin.

 

What is the opposite of fair? Something is unfair if it favours one party over another.

 

BIASED

If a coin tends to come up heads more often than tails, it is biased.

 

A chance setup is unbiased if and only if the relative frequency in the long run of each outcome is equal to that of any other.

 

INDEPENDENCE

 

Lack of bias in the system does not guarantee that a chance set up is fair. We need more than that. The idea of a fair tossing device seems to involve there being no regularity in the outcomes. Or they should be random. Randomness is a very hard idea. Outcomes from a chance setup are random (we think) if outcomes are not influenced by the outcomes of previous trials.

 

The idea of complexity is also used. Random sequences are so complex that we cannot predict them. The complexity of a sequence can be measured by the length of the shortest computer program needed to generate the sequence. A sequence is called random, relative to some computational system, if the shortest program needed to generate it is as long as the sequence itself.

Here we have a family of related ideas:

Random: no influence from previous trials.

No regularity: no memory of previous trials.

Complexity: impossibility of a gambling system.

 

TWO WAYS TO BE UNFAIR

 

Hence a chance setup can be “unfair” in two different ways. It can be biased. For example, heads tend to come up more often than tails. But there could also be some regularity in the sequence of outcomes. Trials may not be independent of each other. Since there are two ways to be unfair, there are four possible combinations:

 

Fair: unbiased, independent.

Unfair: unbiased, not independent.

Unfair: biased, independent.

Unfair: biased, not independent.

 

The Gambler’s fallacy:

What is the gambler’s fallacy? The fallacy does not involve bias. It involves independence.

The gambler thinks that a sequence of twelve blacks makes it more likely that the wheel will stop at red next time. If so, a past sequence affects future outcomes. So trials on the device would not be independent, and the device would not be fair after all.

 

IMPOSSIBILITY OF A SUCCESSFUL GAMBLING SYSTEM

 

There are many ways to think about randomness and independence. One definition says that outcomes are random if and only if a successful gambling system is impossible. That does not mean that you can’t win. Someone has to win. A gambling system is impossible if no betting system is guaranteed to win.

 

Our fallacious gambler dreams that a profitable gambling system is possible. His system is, in part, “When you see 12 blacks in a row, bet red.” This system would work only if the spins of the roulette wheel were not independent. And,  ofcourse, they may not be! But if the gambler’s premise is that the roulette wheel is fair, then he should think the spins are independent of each other. So his fallacy is thinking both that the roulette wheel is fair, and  that a gambling system is possible.

 

AN ODD QUESTION

 

In Lotto 6/49, a standard government-run lottery, you choose 6 out of 49 numbers (1 through 49). You win the biggest prize-maybe millions of dollars-if these 6 are drawn. (The prize money is divided between all those who choose the lucky numbers. If no one wins, then most of the prize money is put back into next week’s lottery.)

 

The very large prizes are split between all the people who choose the lucky numbers that win. It may be that most people like irregular-looking outcomes like B. They cannot believe that a regular sequence would occur. So fewer people choose a regular sequence than choose an irregular sequence.  Hence if a regular sequence is drawn, the prize may be bigger, for each winner, than if an irregular sequence is drawn.

 

But! Maybe enough people know this so that they try to outwit thew herd, and become a little herd themselves. If a regular sequence had come up (it did not) the payoff would have been quite small, because the prize would have been split among so many players.

 

ANOTHER ODD QUESTION.

 

Hardly anyone making up a sequence of 10 tosses puts in a run of seven heads in a row. It is true that the chance of getting 7 heads in a row, with a fair coin, is only 1/64 (½ X ½ X ½ X ½ X ½ X ½ X ½). But in tossing a coin 100 times, you have atleast 93 chances to start tossing 7 heads in a row, because each of the first 93 tosses could begin a run of 7.

 

It is more probable than not, in 100 tosses, that you will get 7 heads in a row. It is certainly more probable than not, that you will get at least 6 heads in a row. Yet almost no one writes down a pretend sequence, in which there are even 6 heads in a row.

 

This example may help some people with the gambler’s fallacy. We have this feeling that if there have been 12 blacks in a row, then the roulette wheel better hurry up and stop at red! Not so. There is, so to speak, all the time in the world for things to even out.

 

THE ALERT LEARNER

We begin with a person we call Fallacious Gambler. He meets Stodgy Logic. Stodgy Logic says to Fallacious Gambler:

 

Your premise was that the setup is fair. But now you reason as if the setup is not fair! You think that the only ingredient in unfairness is bias. You forgot about independence. That is your fallacy.

They meet a third party Alert Learner. She reasons:

We’ve been spinning black with this wheel all too frequently. I thought that the wheel was unbiased. But that must be wrong. The wheel must be biased toward Black! So I’ll start betting on Black.

 

She has made an inductive argument: to the risky conclusion that the wheel I biased. Alert Learner bets black. Fallacious Gambler bets red. Stodgy Logic there is no point in betting either way. Who is right? Alert Laerner might be right. Maybe the wheel is biased. Pure Logic cannot tell you.

 

RISKY AIRPLANES

People argue differently. Each brings in different premises. A premise may be false. Logic cannot tell which premises are true. Logic can only tell when premises are good reasons for a conclusions.

 

MODELS

 

In reasoning using probability, we often turn to simple and artificial models of complex situations. Real life is almost always complicated, expect when we have deliberately made an apparatus and rules-as in a gambling situation. People made the roulette wheel to have some symmetries, so that each segment turns up as often as every other. Real life is not like that. But artificial games may still be useful probability models that help us to think about the real world.

 

TWO WAYS TO GO WRONG WITH A MODEL.

 

Serious thinking about risks, which uses probability models, can go wrong in two very different ways

• The  model may not represent reality well. That is a mistake about the real world.
• We can draw wrong conclusions from the model. That is a logical error.

 

Two kinds of error recall the two ways to criticize an argument.

• Challenge the premises-show that at least one is false.
• Challenge the reasoning-show that the premises are not a good reason for the conclusion.

Criticising the model is like challenging the premises. Criticising the analysis of the model is like challenging the reasoning.