The “Gambler’s Fallacy”

In previous newsletters, Clark & John’s probability whizz-kid Rob Clark, informed us of the actual chances of winning a prize on the UK National Lottery and also entertained readers with a fascinating article on the likelihood of multiple birthdays.

For this issue, Rob looks at the “Gambler’s Fallacy”.

We hear phrases all the time don’t we, like ….”it’s got a 50% chance of occurring.”
The mistake most people make, is that they take the statement literally.

When reading the statement above, you’d assume that if tossing a coin, say 10 times, they’ll automatically be 5 heads and 5 tails as a result, as each scenario has indeed a 50% chance of occurring (coin landing on it’s side is ignored). Most people’s expectations are on too narrow a scale. The assumption above, of 5 heads and 5 tails, ultimately leads to the wrong assumption of 7 heads out of 14 tosses and 15 heads out of 30. This is of course untrue and should not be assumed. The laws of probability are not that precise. This perception that these expectations will be realised of a short time is known as the Gambler’s Fallacy. It is this same fallacy that causes roulette players to assume that red is more likely to win after a run of consecutive black wins. Roulette wheels have no memory and it’s simply coincidence. The mathematical law that dictates that your expected incidence of wins will match that of your expectation is effective ONLY when dealing with a large amount of incidents, or in the gambler’s case, bets.

It is referred to as the “Law of Large Numbers” and this example my help your understanding…..

Let’s imagine you’re diving along the motorway, behind a lorry that’s carrying a crate of 10,000 ping pong balls. There are say 6,000 red balls and 4,000 white balls and they have been thoroughly and randomly mixed up in the crate. Suddenly, the crate shatters in front of your car, sending all 10,000 ping pong balls bouncing randomly onto the motorway. Let’s say you happen to run over 10 of these balls.

How many red balls do you smash?

Following our fallacy above, you’d probably say 6, since 60% of all the balls are red and 60% of 10 equals 6.

However, this is only the most LIKELY outcome, of several other likely and not-so likely outcomes. It’s not the only one. You are only slightly LESS likely to smash 5 red balls or even 7 red balls, than you are to hit exactly 6. You’re also only slightly less likely to smash 4 red ones or 8 red ones etc. You are restricted to only 11 different possible outcomes ie.0-1-2-3-4-5-6-7-8-9 or 10 red balls. In fact, if you add up all the chances of all the other combinations of outcomes (smashing anything BUT 6 red balls out of 10), it’s plain to see you are actually not at all likely to smash exactly 6 red balls. Even though you do indeed have a 60% expectation, you will most likely end up with some other combination.

The obvious reason for this is that 10 observations is simply too few, from which to expect predictable results. This is at the heart of the Gambler’s Fallacy and as the name suggests, has been the undoing of many a gambler!

Let’s take another look at the moment the crate shatters and suppose this time you smash 1000 balls. Now we have a wholly different proposition. In this scenario and if you were a gambler, you could safely bet all your savings that you will smash more red than white balls and in fact, can be reasonably certain that you’ll smash between 570 and 630 red balls! (Remember, the bigger the numbers, the more predictable the outcome).

Without saturating you with too much mathematical proof, with this size of “sample” you might not smash 60% but still smash 59.9%, or 60.1%, or 59.8% or 60.2% etc. With 1000 balls smashed, you might not hit exactly the 60% expected figure (red balls) but you could not miss your expectation by as much as 10%. It would take a mathematical long-shot of miraculous proportions to smash say only 500 red balls, or as many as 700 red balls. The reason for this…..there are simply too many more events likely to occur (ie smashing 599 balls or 601, or 598, or 602 etc.)

As is so often the case, the “obvious” answer to the question posed above is not always correct and just a small amount of thought and examination can lead to the perhaps “unusual” but correct answer. (See the last Newsletter article on birthdays for an excellent example of this).

So be careful driving on the motorways this Christmas and if you do take to the road, make sure you’re not too close to the lorry in front carrying pin pong balls………

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