The 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 that 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.

An update:

I did not realize that many mathematicians were puzzled by this paradox, until I saw it on wikipedia at: https://en.wikipedia.org/wiki/Two_envelopes_problem

So, two years after publishing the above explanation, I decided to put it in simple algebraic form.

We have been given an envelope, and it contains the amount x.

This implies that the predetermined, but unknown total for the two envelopes is either x + x/2, or x + 2x.

First scenario: The total, for the two envelopes, is x + x/2.

Case 1: Our envelope contains the larger amount x.
By switching we lose the difference between the two: x/2.

Case 2: Our envelope contains the smaller amount x/2.
By switching we gain the difference between the two: x/2.

Second scenario: The unknown total, for the two envelopes, is x + 2x.

Case 1: Our envelope contains the larger amount 2x. By switching we lose x.

Case 2: Our envelope contains the smaller amount x. By switching we gain x.

Thus, in both scenarios, the probability of losing or gaining is the same.

What about a larger difference?

The same is true even if one envelope contained 100 times the amount of the other.

Then the total would be either x + x/100, or x + 100x

First scenario: The total, for the two envelopes, is x + x/100.

By switching we may equally lose, or gain, the same amount: the difference between the two envelopes: x-x/100.

Second scenario: The total, for the two envelopes, is x + 100x.

By switching we may equally lose, or gain, the same amount: the difference between the two envelopes: 100x-x.

Thus, in both scenarios, the probability of losing or gaining is the same.

A more general formula

So, no matter what the multiplier m is, the unknown amount of the total for the two envelopes can be expressed as either x + x/m, or x + xm.

In the first scenario, by switching envelopes we may lose, or gain, the same amount: x — x/m.

In the second scenario, by switching envelopes we may lose, or gain, the same amount: xm — x.

The Trigger

What triggers the paradox is that some information is given about the relation between the two envelopes, without providing the essential information: The total amount. Thus, if we eliminate the misleading information, the paradox does not materialize.

It is intuitive to see that we cannot enhance our probability of getting a larger amount by switching envelopes, when the envelopes contain random amounts: x and y. Then, the total for the two envelopes can be expressed as: x + y.

Both scenarios are analogous, whether x is greater than y, or y is greater than x. By switching envelopes it would be equally probable to lose, or gain, the difference between the two amounts, no matter what the difference is.