Post by Erasmus on Mar 28, 2011 17:33:11 GMT -5
This is a well-known paradox of probability often used to show the mistakes of intuition and 'common sense'. I think it shows the mistakes of trusting in numbers instead of intuition and common sense.
There was a quiz show of the 'choose the box' type with three doors. Behind one is a car, behind the others goats (or Dusty Bins - Gods that was an awful show!).
You are asked to choose a door. The host then opens one of the doors to reveal a goat and asks if you want to change your mind.
On the face of it, there is no reason to do so because you know you either have the car or the goat and that' exactly what you had before. However, statistics says otherwise. On your first choice there was one chance in three of choosing the car. Once the goat is revealed, there is one chance in two, so changing your mind changed your chances from 33% to 50% - counter-intuitive.
I dispute this. I don't dispute the figures, they are simple enough. I dispute their usage. It's easy enough to demonstrate. The official line runs that you had 1:3 chance of choosing the door you did, but when you change your mind after the revelation you have 1:2. Fair enough, but if you now have 50% chance of getting the car if you do choose the other door, it follows that you have the same 50% if you don't. You can't have 33% chance and 50% chance out of two options!
What has happened is that the first choice actually has no bearing on the result. In a sense, it is a kind of mental illusionist trick diverting attention away from the real issue. It deliberately confuses statistical analysis of subjective knowledge with objective reality and presumes that the numbers cannot change.
Objective reality is 1:3 chance that the car is behind any one door. At the start, your subjective knowledge is the same. That is not true for the studio staff though: they know exactly where the car is! so their subjective knowledge is 100% correct.
Whatever you choose first, the host is going to open a door with a goat behind it. Therefore that door is excluded from your initial choice. If it will be eliminated then it has been eliminated. Subjective knowledge may well be described as 1:3 but objective reality is 1:2 - the car is behind one of two doors in the final revelation, the other door being irrelevant. Your actual chances were always 1:2.
It becomes obvious if it is always the same door that is opened. Obviously that door is not part of the equation. In a case like that subjective knowledge and objective reality coincide. Not knowing which specific door will open does not alter the fact that the final choice is between a car and a goat and that is all that matters.
There are similarities to the gambler's confusion that the longer a number does not come up, the greater its chance of doing so. In this case it is the other way round! If you are dealing a pack of cards where a card can only come up once, then increasing odds is true, but for a roulette wheel or if the card is returned to the pack, probability over time is not at all the same thing as probability per trial.
For all you know, your number does not come up because the roulette is fixed or because what you thought was a random number generator is in fact a complicated routine that the longer it continues makes numbers more likely to repeat and lowers the chances on those that have not come up. (These are exasperatingly easy to produce when trying to create elaborate random generators)
There was a quiz show of the 'choose the box' type with three doors. Behind one is a car, behind the others goats (or Dusty Bins - Gods that was an awful show!).
You are asked to choose a door. The host then opens one of the doors to reveal a goat and asks if you want to change your mind.
On the face of it, there is no reason to do so because you know you either have the car or the goat and that' exactly what you had before. However, statistics says otherwise. On your first choice there was one chance in three of choosing the car. Once the goat is revealed, there is one chance in two, so changing your mind changed your chances from 33% to 50% - counter-intuitive.
I dispute this. I don't dispute the figures, they are simple enough. I dispute their usage. It's easy enough to demonstrate. The official line runs that you had 1:3 chance of choosing the door you did, but when you change your mind after the revelation you have 1:2. Fair enough, but if you now have 50% chance of getting the car if you do choose the other door, it follows that you have the same 50% if you don't. You can't have 33% chance and 50% chance out of two options!
What has happened is that the first choice actually has no bearing on the result. In a sense, it is a kind of mental illusionist trick diverting attention away from the real issue. It deliberately confuses statistical analysis of subjective knowledge with objective reality and presumes that the numbers cannot change.
Objective reality is 1:3 chance that the car is behind any one door. At the start, your subjective knowledge is the same. That is not true for the studio staff though: they know exactly where the car is! so their subjective knowledge is 100% correct.
Whatever you choose first, the host is going to open a door with a goat behind it. Therefore that door is excluded from your initial choice. If it will be eliminated then it has been eliminated. Subjective knowledge may well be described as 1:3 but objective reality is 1:2 - the car is behind one of two doors in the final revelation, the other door being irrelevant. Your actual chances were always 1:2.
It becomes obvious if it is always the same door that is opened. Obviously that door is not part of the equation. In a case like that subjective knowledge and objective reality coincide. Not knowing which specific door will open does not alter the fact that the final choice is between a car and a goat and that is all that matters.
There are similarities to the gambler's confusion that the longer a number does not come up, the greater its chance of doing so. In this case it is the other way round! If you are dealing a pack of cards where a card can only come up once, then increasing odds is true, but for a roulette wheel or if the card is returned to the pack, probability over time is not at all the same thing as probability per trial.
For all you know, your number does not come up because the roulette is fixed or because what you thought was a random number generator is in fact a complicated routine that the longer it continues makes numbers more likely to repeat and lowers the chances on those that have not come up. (These are exasperatingly easy to produce when trying to create elaborate random generators)
