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DOUBLE OR HALF
Here is an amzing paradox.
A friend hands you an unmarked envelope (E1) and tells you that it contains a number of $100 bills. You are about to open it, pleased with your friend's generosity, when he produces a second envelope (E2). He then tells you that E2 contains either haff or double the amount contained in E1. Furthermore, he gives you the choice of sticking or swapping. What would you do?
You would probably switch, arguing as follows:
Suppose E1 contains $1,000. By definition, E2 contains either $2,000 or $500. In other words, I stand to gain $1,000 dollars but risk losing $500. As the odds are equal, I should swap.
Is this line of reasoning correct?
When I read the puzzle, I can find no fault with the above reasoning and the solution confused me. Here is the solution again exactly as written from the book:
No. Surprisingly, it makes no difference whether you switch or stick. Here is the proof:
Suppose you have twenty envelopes on the table, ten of which contain $1,000 each and the other ten contain $2,000 each. This arrangement satisfies the conditions of the paradox, inasmuch as if you pick an envelope and then switch, you could either double or halve the your money. Yet, once you pick ten envelopes , whether you switch or stick, your average take will be $15,000.
The essential difference between a decision based on your reasoning and the above proof is that the envelopes are unmarked. If for instance, E1 were marked $1,000 and you would then know that E2 contained $2,000 or $500, you should switch.
I have read and reread it, and thought it over again and again and am still confused. I do not understand how knowing the amount of money in E1 affects the decision. Lets say that amount of money in E1=X. This would mean the amount of money in E2 either equals 2X or .5X. In the long run, half the time E2 equals 2X the other half it equals .5X. Now, lets do this the way book does. If I decide not to switch every time, I end up with 20X, but if I decide to switch every time, I end up with 25X. I can not figure up how my reasoning is wrong and the book's is corret. Is there anyone here who is better at these puzzles than I am? I know that is stupid to let a puzzle annoy you like this, but puzzles like this get under my skin. Can anyone help?
Yes I know, I am a nerd/geek and have no life.