Jesse Mazer wrote:
I don't think that's a good counterargument, because the whole concept of probability is based on ignorance...
No, I don't agree! Probability is based in a sense on ignorance, but you must make full use of such information as you do have. If you toss a fair coin, is Pr(heads)=0.5? According to your argument, it could actually be anything between zero and one, because it is possible I am lying about it being a fair coin!
Here is another "two envelope" example:
Two envelopes, A and B, contain two doses of the drug Lifesavium, the Correct Dose and the Half Dose. If you give the patient more than 1.5 times the Correct Dose you will certainly kill him, while if you give him the Half Dose you will save his life, although he won't make an immediate recovery as he would if you gave him the Correct Dose. If you don't give him any medication at all, again, he will surely die. Once you open an envelope, the medication in in such a form that you must give the full dose, or nothing.
You are faced with the two envelopes, the above information and the sick patient, with no other help, on a desert island. There is one further complication: if you open the first envelope, and then decide to open the second envelope, you must destroy the contents of the first envelope in order to get to the second envelope.
OK: so you open envelope A and find that it contains 10mg of Lifesavium. You don't know whether this is The Correct Dose or the Half Dose; so envelope B may have either 5mg or 20mg, right? And if 10mg is the Correct Dose, then if you discard envelope A in favour of envelope B, there is a 50% chance that envelope B will have double the Correct Dose and you will kill the patient - so you had better stick with envelope A, right?
I think you can see the error in the above argument. You already know that the amount in each envelope is fixed, so even though you have no idea of the actual dosages involved, or which envelope contains which dose, even after opening the first envelope, there is NO WAY you can give the patient an overdose. There is no way envelope B can contain 20mg of Lifesavium, but even though you cannot know this, you can use the above reasoning to deduce that there is no expected benefit from choosing a strategy of switching or not switching - as you can also see intuitively from the symmetry of the situation, whether you choose envelope A or B first.
In the game with the envelopes and the money, the analogous error is to think that there is a possibility of doubling your money when you have actually picked the envelope containing the larger sum first. As I said in my previous post, if this assumption is valid, then you are playing a different game in which our eccentric millionaire flips a coin to decide (without telling you which) if he will put double or half the sum you find on opening envelope A into envelope B. You would then certainly be better off, on average, if you switched envelopes.
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