Originally Posted by
Three
The guy proves the probability that you picked the envelop with the lower amount is 50%. That never changes with no information on each envelop. It still doesn't change when you know what is in one envelope. You still have a probability of 50% that you chose the larger amount. But the symmetry that creates the paradox gets destroyed once you know the amount in one envelope. If this guy thinks he proved that after finding out the amount in on envelope that the decision on which envelope to choose isn't clear he is an idiot. Know what you have to watch out for is the person running the show can choose whether to show you the higher amount or the lower amount. If the amount shown was chosen at random and is equally likely to be the larger amount as the smaller amount choosing the unknown envelope is better.
From the link in bold:
1) When you make your choice (C)
2) Either the envelope containing the lower value (C=lower) or the one that contains the higher (C=higher).
3) During the game, the value (V) of the content of the chosen envelope is revealed to be a certain value M.
The paradox above arose because you assumed that
4) P(C=lower|V=M) = P(C=higher|V=M) = 0.5
No, the paradox arose because M was a variable. You make it a constant then V is the same constant. That is your number 3 in the bold.
Let's see why this cannot be the case. (In what follows, remember that L, V, C are variables and M is a numerical constant.)
No, V is not a variable. If V=M and M is a constant then V is also a constant. You can't treat it like a variable. See your number 3 above. C is a constant as well. C is the envelope you chose. If you know nothing about the contents of either envelope you can view it as a variable but once you know the amount in one envelope it is no longer a variable. It is the envelope that contains the amount M which is a constant. It's value, V, is equal to M which is a constant. So L is your only variable. And you have 3 constants; C,V, and M where V=M.
The paradox above arose because you assumed that
P(C=lower|V=M) = P(C=higher|V=M) = 0.5
No, that was a given. The paradox arose because M was a variable.
Let's see why this cannot be the case. (In what follows, remember that L, V, C are variables and M is a numerical constant.)
No, L is the only variable. We Know M is a constant. V=M so V is also a constant. And C is the envelope you chose which contains the amount M, a constant and has the value, V, also a constant.
P(C=lower|V=M) = P(C=lower)P(L=M)/P(V=M) . . . (2)
and P(C=higher|V=M) = P(C=higher)P(L=M/2)/P(V=M) . . . (3)
I skipped to this because with all the constants the preceding was pretty meaningless.
Given V=M, (P(V=M)=1), this reduces to P(C=lower|V=M) = P(C=lower)P(L=M) and P(C=higher|V=M) = P(C=higher)P(L=M/2). Both terms mean the exact same thing so what is the point both statements are givens.
From (2), P(C=lower|V=M) is the same as P(C=lower) only if P(L=M) is the same as P(V=M).
No. The other possibility is the probability that V=M is 1. V=M is 100%, that was how you defined M and V. M is a constant that is the amount in the envelope. V is the value of the contents of the chosen envelope which is the constant M. So the probability that V=M is 1. So P(L=M) does not have to be the same as P(V=M). Your treating constants as variables lead to this false statement. Everything based on this falsehood is wrong.
I can't believe this guy thought he could prove that the odds you chose the envelope with the larger amount or smaller amount can't be equal at .5 and .5 LoL
The paradox above arose because you assumed that
P(C=lower|V=M) = P(C=higher|V=M) = 0.5
Let's see why this cannot be the case.
Well I guess if you take a given like V=M, and then define V as a variable and M as a constant you can make it look like you did to those that don't try to follow what you did.
(In what follows, remember that L, V, C are variables and M is a numerical constant.)
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