The Two Envelope Paradox

 

There are two envelopes containing money. One has twice as much money as the other. You choose an envelope, you open it, and can keep it, or swap it for the other. Should you keep it or switch?

Method 1. Let the envelopes contain x and 2x. If you find x in your envelope then by switching to the other envelope you gain x. If you find 2x then switching loses x. The net gain is zero. Therefore switching is of no benefit.

Method 2. If your envelope contains x then the other one must have 2x or x/2. Switching means either a gain of x or a loss of x/2. Since these are equally probable you should switch.

Say you choose an envelope and find it contains $100. Should you keep it or switch? If you switch you will either gain $100 or lose $50. These are equally likely, so switching makes sense. Or does it?

Swapping sounds a good idea but leads to absurdity. This is because discovering that the first envelope contained $100 is irrelevant. Whatever that envelope was found to contain, whether $1, $10 or $770, exactly the same argument would lead us to switch. Thus if we found $x in the first then switching promises either $2x or $x/2, ie a gain of $x or a loss of $x/2. This means, statistically speaking, a gain if we swap for any value of x. The same argument can be used equally well for the second envelope before we open the first one. Hence we would have been compelled to switch from the second envelope to the first, had we picked the other one initially.

When I first saw this puzzle, I thought I would knock it over in five minutes. Eight weeks later I was still agonising over it. I looked at more than 40 answers on the Net but found that they were either too hard for me to follow, inadequate or plain wrong. Eventually, using some useful insights from the Net, I came up with a solution which I believe is correct. However, my solution is absurdly long and complicated for what appears to be a trivial problem.

My solution

If you want to avoid reading a mathematical solution then try the Simple Explanation

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