A Simple Solution of the Envelope Exchange Paradox

We are shown two envelopes: One envelope contains twice as much money as the other.

Having chosen one envelope, before inspecting it, we are given a chance to take the other envelope instead.

It would appear that it is to our advantage to switch envelopes if we reason as follows:

1. Let’s suppose that the amount in our selected envelope is $20.

2. The probability that it is the smaller amount is 50%, and that it is the larger amount is also 50%.

3. The other envelope may contain either twice as much ($40) or half as much ($10).

4. If we switch envelope, then we could gain $20, or lose $10. Thus we are better off by switching envelopes.

We know that the above reasoning is wrong, because after switching envelopes we could use the same reasoning to switch the envelopes again!

So, how do we solve this apparent paradox?

You may want to think about it, before reading any further.

A Simple Solution:

We do not know the amount in our selected envelope. However, we know that the total amount in both envelopes is fixed and pre-established.

We can suppose that our selected envelope contains any amount (e.g. $20).

This implies the total amount for the two envelopes would be either ($20 + $10 = $30) or ($20 + $40 = $60).

The key point is that the two totals cannot be true at the same time.

Thus each hypothetical scenario must be analyzed separately, and not as in points 3. and 4. above.

In the first scenario (Total = $30), our supposition that our selected envelope contains $20 would imply that the other envelope contains $10. However, it is equally probable that the opposite is true: our selected envelope could contain only $10 and the other $20.

In both cases, by switching the envelope, we may gain or we may lose $10.

Similarly, in the second scenario (Total = $60), our supposition that our selected envelope contains $20 would imply that the other envelope contains $40. However, it is equally probable that the opposite is true: our selected envelope could contain $40 and the other $20.

In both cases, by switching the envelope, we may gain or we may lose $20.

Thus, in both scenarios, there is no probabilistic advantage in switching envelopes.