Monday, January 12, 2009

Monty Hall

A friend outlined an interesting problem that shows that intuition cannot be relied upon. It is a maths problem rather than a philosophical one but I have extended the problem to an epistemological one.

The original problem (see internet for more details) goes something like this.

You are in a game show and there are three doors in front of you. One of them hides a car and the other two have $1 dollar bills behind them. The game show host asks you to pick one of the doors (hence the Monty Hall title of the problem). You pick one of the doors and at this point you have a 1 in 3 chance of getting the door with the car behind it. Monty, knowing which door holds which prize, opens up one of the doors with the $1 dollar bill behind it leaving two doors still closed, one with the car, one with the remaining dollar.

The question is, would you change your selection from the first door you picked to the now remaining other door or remain with your selection? Don't forget, you have 2 doors left closed one of which contains the car.

The answer is one which has had the mathematical world in turmoil. You should change your selection away from the one you originally picked. Why? Your intuition problem leads you, like most others, to think that there are two doors and only one with the car so the chances of you having the one with the car are 1 in 2 (or 50/50). Unfortunately you would be wrong. The odds of your original door winning are still 1/3 and so the odds of the other door being the car are actually 2 in 3 or 66.66%.

This took a lot of convincing for me but I have seen the spreadsheet and proof that this is actually the case. But that is just maths. I extended the problem for my friend to make it a philosophical one. Here's how.

You have the two doors left, one with the car and one with the dollar bill. You now know (subjecting to accepting the maths) that your best bet is to change your door choice to the other remaining door. And 66.66% of the time you would win the car.

BUT, I will now walk you out of the studio and I will take a stranger off the street and tell them nothing of what has gone on. I lead him into the studio and show him the two doors (for sake of argument I have taken away the open door so only two doors remain). One contains the car and one a dollar bill. He knows nothing of what has gone on. Monty asks him to pick a door.

Now the interesting thing is that the doors are exactly the same as they were so what are the odds of the stranger picking the car. The odds must be 50/50. There are 2 doors behind them and only one has the car.

So for one person with knowledge of the 3rd door they had a 66.66% chance of winning (proven) if they picked the remaining door but the stranger without this knowledge only has a 1/2 chance. If we ran the test with the stranger they would win 50% of the time.

It is hard to swallow as it implies (to me) that knowledge somehow alters the odds from 2/3 to 1/2 chance. The interplay of odds and knowledge and how the physical results actually match are strange. The original point I was making was that none of this is intuitive so how well can we rely upon intuition?

We can take this a step further. What are we actually saying when we calculate probability. There are two ways, IMHO, that we can look at it.

1) Probability is the odds that any given physical event will occur.
2) Probability is the odds that we will guess the right outcome of any physical event. I.e. it is an epistemological issue.

I vote the later. The Monty Hall problem I think is explained much better when we think of it as 2) and not 1).

The second reason is determinism. If you have a 6 sided die then the probability of the number scored when the dice is rolled is 1 and the probability of the dice landing on any other face is actually 0. In other words, when that dice is rolled it cannot end up on any other face than which it actually does. So given the future landing state of the dice can only ever have landed as it did the physical probability of it was certainty. It could not have been any other way.

The only probability of interest is then type 2). That we don't KNOW what that certain outcome will be so we guess. So probability is about knowledge, or lack of it, rather than to do with anything physical.