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Saturday, April 25, 2009

A common fallacy

(This is an English translated version of my previous post)

Our brain sure is impressive. It is capable of maths, logic, problemsolving, spatial awareness in addition to all those small things it does for us automatically (like, y'know, controlling our body...)
The way we use our brain though, is far too often not impressive. We will fall naturally into fallacious thinking and wrong judgments. This of course comes from our evolutionary background - we need fast judgments to survive. To better avoid predators and other dangers, we have a strong tendency to see patterns, but this often leads to us seeing patterns and shapes where there in reality are none.

In this series, I would like to shed some light on some common errors that most of us do fairly often. I would not claim myself to be immune to these - but if we are more aware of them then perhaps we can avoid them more often, which would be better.

Probabilities

Of all the things that our brain does correctly, estimating probabilities is not one of them. In fact, we downright suck at it. We like being cautious and prepared, but at the same time we grossly misrepresent chances at everything involving ourselves. Let me use the norwegian lottery "Lotto" as an example.

Norsk Tipping has this to say about the chances of winning: (translated)

It isn't easy to become a millionaire through Lotto but it is also not easy to say exactly how hard it is. At least not in practice. In theory one can say that it is almost impossibe to get seven correct numbers, but on the other hand on average four people manage this every saturday.

Already here we can see how hard it is to think about probabilities - and that is before we have even seen a single number! It is "almost impossible" but at the same time "four people manage this every saturday." Our brain does not have any problems reconciling these two statements. There is also present in the paragraph a clear theory-practice difference - in theory it is impossible but in practice it might happen. So far still good for our brains even if with a bit of proper logical thinking you might see that these statements are in fact contradictory.

Let's instead move onto looking at the exact numbers and chances to win. Lotto has 34 numbers, of which you have to correctly pick seven. According to Norsk Tipping the chance of hitting that is 1:5.379.616. One out of every five million four hundred thousand, rounded. The main thing here is that no matter how you try to illustrate or visualise it, it is such a large number (or to be more correct, such a small number) that our brain cannot visualise or imagine it. And then our brain falls back to overestimating just to be on the safe side.

Norway has a little less than five million inhabitants. Thus the chance of actually winning would be about the same as to take a random phone book, open it to a random page, random column, and point to a random row and point at your own name. (Strictly speaking you have a greater chance of pulling off this feat, as we are not 5.4 million only 4.8 million still... but the point stands)

And yet, people play Lotto. And yet, people win. Every week, someone wins. (This is not an anti-Lotto post. I am quite proud of my 530% return on investment over my Lotto career)1
The odds are still astronomical. With enough tries, even the most astronomical odds can be overcome. It's very easy to fall for the fallacy "It has happened to someone therefore it can (will) happen to me." It can happen but probably not. And that is where most of us trip and fall.

Other examples of where our way of dealing with probabilities messes up our brain:
  • The Birthday Paradox, or, if you have gathered 23 random people then the chance of two of them sharing a birthday is 50%. The reason is that we think "Oh, there are 365 days in a year so the chance of two people sharing a birthday is 1/365" and then multiply that by 23 giving 23/365. This however is the wrong way of working out the probability. In reality the probability is 23*(22/2) since you have quite a number of pairs that can match. In a group of 23 people there are 253 distinct combinations of people that can be compared!
  • We give higher probabilities to things we recently have thought about. Airplanes and airtravel is astoundingly safe and reliable, thanks to all the work the airlines put into making them safe and reliable. However just after an airplane accident (they do happen, just quite rarely) air flights take a noticeable dip, even if the chance of another crash happening is not corellated with the first one. On the contrary, the argument could be made that due to the extra awareness of the flight crews following another accident, you are less prone to experiencing an incident, but that does not stop us from not flying after a crash.
  • Likewise, the chance of you being involved in an automotive accident dwarfs the chance of you crashing in a plane. But still, we worry excessively about one and not the other, when it should be reverse.
  • Hollywood does not help the cause either. In all movies and TV-show the most improbable things will happen with incredible frequency. Of course, it is understandable, a movie would be quite boring if only things that were statistically likely happened. Imagine a movie about an airplane that does not get hijacked, a bank that does not get robbed, or a statesman who does not get assassinated. Quite dull in that case, but it is the norm for every day for the rest of us. Still, it poisons the mind in such a way that we can get used to expecting the incredulous to happen.
This is of course just a very small extract of the events where we can quite simply be wrong when it comes to probability and calculations. We quite simply suck at it. Most of what we can hope to do is to know that we are bad at it, so we can try to adjust our initial estimates in the correct direction.



1At age 12, after my horoscope recommended it, I played Lotto. I wagered 10 Kroner (bought five sets of seven numbers) and won 53 kroner from it. I have never gambled since.

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