Tilted Forum Project Discussion Community - View Single Post - Another of those crazy paradoxes (similar to Monty Hall problem)

KnifeMissile · 12-11-2003, 05:42 PM

Both evaluations of expectation are correct. I think the problem is simply that you're taking expectation too seriously. First, I will try to show how the two evaluations are correct, since there seems to be some confusion about this, and then I'll try to show why they are not contradictory.

First, suppose we play a game. I will give you some amount of money and show you a box. In the box is either half the amount of money I've just given you or double it. Given a choice between the two, which should you take? Expectation says you should always pick the other box and, in this case, it will be correct! If you were to do this several times and always pick the sure thing then, on average, you will find yourself with less money than if you had always picked the other box. This is reflected by the expectation being 1.25 * X, where X is the amount of money you already have.
By the way, this is precisely the game you play after picking the first cup of this thread's game. In this case, X is the amount of money revealed. Please note that you do not know weather X is the greater or lesser money.

Secondly, lets play this thread's game. It would seem that it doesn't matter which cup you choose since you don't know any difference between the cups. If X amount of money is in one cup and 2 * X is in the other then the expectation is 1.5 * X. Please note that, in this case, you know that X is the lesser money.

X is not the same value in both cases so you can't simply compare them. The evaluations of expectation are correct, it's just that the expectation doesn't suggest what you think it does because they're both different situations.

I've always liked this story when trying to explain probability. I'm going to the corner store and I'm definitely buying a tub of ice cream. They only carry the two flavours chocolate chocolate chip and coockie dough. What are the chances that I'm buying the cookie dough?

Well, if this is all you know, then the chances are 50%.

But, if you happen to know that chocolate chocolate chip out-sells cookie dough, say by 9:1, then you can say that the chances I'll buy the cookie dough are 10%.

But, if you happen to know that my favourite flavour is cookie dough and I buy it more often than chocolate chocolate chip, say by 9:1, then you can say the chances I'll buy the cookie dough are 90%.

Which probability is correct? They all are! Probability isn't an absolute, it depends on what you know. Even when you're talking about seemingly perfectly random events, like dice, you still must make the assumption that all the sides of the dice are equally as likely to occur. The dice could have been weighted, or irregular, or anything! Whenever calculating probability, you are always making assumptions because you have insufficient knowledge to be certain...

12-11-2003, 05:42 PM	#7 (permalink)
KnifeMissile Location: Waterloo, Ontario	Both evaluations of expectation are correct. I think the problem is simply that you're taking expectation too seriously. First, I will try to show how the two evaluations are correct, since there seems to be some confusion about this, and then I'll try to show why they are not contradictory. First, suppose we play a game. I will give you some amount of money and show you a box. In the box is either half the amount of money I've just given you or double it. Given a choice between the two, which should you take? Expectation says you should always pick the other box and, in this case, it will be correct! If you were to do this several times and always pick the sure thing then, on average, you will find yourself with less money than if you had always picked the other box. This is reflected by the expectation being 1.25 * X, where X is the amount of money you already have. By the way, this is precisely the game you play after picking the first cup of this thread's game. In this case, X is the amount of money revealed. Please note that you do not know weather X is the greater or lesser money. Secondly, lets play this thread's game. It would seem that it doesn't matter which cup you choose since you don't know any difference between the cups. If X amount of money is in one cup and 2 * X is in the other then the expectation is 1.5 * X. Please note that, in this case, you know that X is the lesser money. X is not the same value in both cases so you can't simply compare them. The evaluations of expectation are correct, it's just that the expectation doesn't suggest what you think it does because they're both different situations. I've always liked this story when trying to explain probability. I'm going to the corner store and I'm definitely buying a tub of ice cream. They only carry the two flavours chocolate chocolate chip and coockie dough. What are the chances that I'm buying the cookie dough? Well, if this is all you know, then the chances are 50%. But, if you happen to know that chocolate chocolate chip out-sells cookie dough, say by 9:1, then you can say that the chances I'll buy the cookie dough are 10%. But, if you happen to know that my favourite flavour is cookie dough and I buy it more often than chocolate chocolate chip, say by 9:1, then you can say the chances I'll buy the cookie dough are 90%. Which probability is correct? They all are! Probability isn't an absolute, it depends on what you know. Even when you're talking about seemingly perfectly random events, like dice, you still must make the assumption that all the sides of the dice are equally as likely to occur. The dice could have been weighted, or irregular, or anything! Whenever calculating probability, you are always making assumptions because you have insufficient knowledge to be certain...