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Human brains weren't made for maths

Updated on July 30, 2014

They are not ?

Human brain is absolutely great at many things. But it often struggles with tasks that have been added to our repertoire since brains evolved. Your computer at home would be hopeless at many things you do easily, but give it a task like finding the square root of 5,181,408,324 and it will have the answer before you even scratched your head (it's 71,982, of course).This just isn't the kind of thing humans were evolved to do - maths doesn't come naturally.

Nowhere is this more obvious that when dealing with probability and statistics. Probability is involved in many of our everyday activities, and statistics are thrown around in the news and politics all the time, yet our brains, developed to deal with images and patterns, have a huge problem dealing with these manipulations of numbers and the impact of chance.

The Monty Hall problem

Let's imagine you're through to the final of a TV game show. The host brings your to the part of the set where there are three doors. Behind two of these doors is a goat, while behind the third door is a grand prize. You want to win the car but don't know which door it is behind. Still, you are asked to pick a door, so you do. There's a one in three chance you have picked the car, and a two in three chance you haven't. Now the host opens one of the doors you didn't pick and show you a goat (he opened the goat intentionally). He now gives you a choice. Would you like to stick with the door you first chose, or switch to the other remaining door?

Now we have one opened door showing a goat, one closed with another goat behind it and the third one closed with prize behind. It seems obvious that there's no difference whether you change your mind or not...

The Monty Hall problem explained by numberphile

What a mess...

The correct answer is: you are better off switching, it's twice as good as sticking. Why? Well think back when you first picked a door. Now let's analyze all possible outcomes using two methods:

  1. You always stick with your door.
  2. You always switch to other door.

Using first method your win rate is 1/3:

  • You chose the door with goat - you lose
  • You chose the door with goat - you lose
  • You chose the door with prize - you win.

Now, what about the second method?

  • You chose the door with goat, second goat is showed by host, you switch to the door with car - you win.
  • You chose the door with goat, second goat is showed by host, you switch to the door with car - you win.
  • You chose the door with car, first goat is exposed by host, you switch from car to second goat - you lose.

You can easily demonstrate that it is better to switch using a computer simulation - it really does work. Nevertheless neither our logic nor intuition prompted that we had at least a slightly bigger chance to win if we changed.

Could you solve these problems without reading this article ?

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The two-boy problem

Again, I'm going to add a simple problem that sounds like this: I have two children. One is a boy. What's the probability that I have two boys? Regular human, who has not acknowledged depths of probability field, in most cases will answer this question as simple as this: 'One's a boy - the other can either be a boy or a girl, so there's a 50:50 chance that the other is a boy.'

Unfortunately that is wrong.

In this particular problem we have 4 variations:

  1. First child - BOY ; Second child - BOY
  2. First child - BOY ; Second child - GIRL
  3. First child - GIRL; Second child - BOY
  4. First child - GIRL ; Second child - GIRL

And we already have a condition - one child is a boy. First three situations fit our condition (at least one boy), but only one of them has what we need - two boys, therefore the probability is 1/3.

Although if we said - 'I have two children. The elder is a boy', only two combinations (1 & 2) would fit this condition because it's particularized (we now know that the first child must be a boy). In this case probability of having two boys would be 1/2.

Now we're equipped to move on to the full version of the problem. 'I have two children. One is a boy born on a Tuesday. What's the probability I have two boys?’ Again, one that hasn't evolved strict mathematical thinking will say: 'The extra information about the day he's born can't make any difference' - and again one would be terribly wrong.

This is how our table should look like

Day of the week
Boy was first born(second child)
Day of the week
Boy was second born(first child)
Tuesday, first child - boy
Tuesday, second child - BOY
Notice that whole Tuesday row is in bold. It's because one combination in this row occurs twice (BOY and BOY). That's why we have to cancel one combination and we are left with 27

We already found the total 27 combinations that fit our 'Boy on Tuesday' condition. Now, which of them are viable for us? We have 14 variants which contain two boys, though two of them are duplicated (both on Tuesday). So we have to cancel one and we are left with our answer: 13/27.

Common sense really revolts at this. By simply saying what day of the week a boy was on, we increase the probability of the other child being a boy. The only way I can think of to describe what's happening is to say that by gaining more information about the boy's birth we can cut out a lot of other options. In other words we are moving towards 50:50 situation - 'the oldest child is a boy'.

Will you survive?

This third example of how bad the brain is at dealing with probability and statistics is one that is much more important for real life.

Let's imagine there's a test for a particular disease that gets the answer correct 95 per cent of the time. Let's say that one in 1 000 people have this disease. And finally a million randomly selected people take the test including you. Now imagine you tested positive in this test. How likely are you to have the disease?

Bearing in mind that the test is 95 per cent accurate, you may well think that you have 95 per cent chance of having the disease, but actually the result is much more encouraging. Actually, since the disease is very rare, most of positive results will come from test's errors. Astonishingly, you would have only 2 per cent probability of actually having the disease even though a great test showed you positive result.

If you are interested in math behind this problem, it would be easier to think that one million randomly picked people (including you) took the test. Every other condition remains. Of those million people tested, 1 000 will have the disease. Of these, 950 will be told correctly that they have it and 50 won't, as the test is 95 per cent correct. 999 000 won't have the disease. Of these, 949 050 will get a correct negative result and 49 500 will get a false positive result. Now of these million people we have 50 450 that heard the positive outcome. You are among them. And only 950 people will actually be infected.

Once more, the way our brains are made simply doesn't fit well with understanding probability.


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