The Exchange Paradox Revisited

September 14, 2026

The Two Envelopes Paradox

In a Nutshell

Imagine it! Two envelopes on the table in front of you, both unopened, both containing cash. One of the envelopes contains exactly twice as much as the other. Call them envelope 1 and envelope 2.

You are called on to open one of them, just one of them. You do so and withdraw £100 (say five £20 notes).

Now, you can keep the £100 or switch it for whatever is in the other envelope.

The question is simple to state. Should you switch? On the face of it, there does seem to be quite a good case for doing so.

The other envelope must contain either £50 or £200, since one of the envelopes has twice in it compared to the other. So, if the other envelope contains £200, switching wins you a net £100. If it contains £50, you lose only £50.

So you have a chance of gaining £100 against an equal chance of losing £50.

That sounds like a good bet.

So it sounds like you should switch. But hold on a bit.

Something has gone wrong

Suppose you do switch. You now have the other envelope. So why don’t we apply the same logic again and swap envelopes once more? Is this also a good bet? After all, if there’s £50 in it, you have an equal chance of winning £100 and of losing £25 by switching again. After all, the other envelope contains either twice as much or half as much. Switching envelopes back and forth suddenly seems like a money making machine. 

Something is clearly wrong.

Start again

Let us suppose the two envelopes contain £100 and £200. If you open the £100 envelope and switch, you will gain £100.

If you open the £200 envelope and switch, you’re down £100. So why switch? Half the time you gain £100 and half the time you lose £100.

So what on earth is going on? Well, when we opened the original envelope and saw £100, we assumed that the other envelope was equally likely to contain £50 or £200.

But was it?

What if we don’t open the envelope?

There is a clever way of seeing the problem.

Choose an envelope but don’t open it for now. Should you switch? There can’t be any reason to do so. You chose randomly between two envelopes and switching simply gives you the one you didn’t choose. There are after all only two envelopes, not three. We imagined £50 in one envelope, £100 in the other, and £200 in the third. In that case, finding £100 in the first envelope means that you can win £100 or lose £50. But that’s in a world of three envelopes. In the real world, there are only two. In our example, they contain either £50 and £100 or £100 and £200, and you don’t know which. They never contain £50 and £100 and £200. And once we have seen £100, we have no reason to assume that £50 and £200 are equally likely unless we know something about how the amounts were chosen.

So should you switch if you find £100? Maybe. If you have some information about how the amounts were selected, £100 might tell you something useful. There could then be a perfectly good reason to switch—or not to.

But if all you know is that one envelope contains twice as much as the other, and there are only two envelopes on the table, not three, there is no general reason to switch.

The strange thing is how convincing the argument for switching seems when you imagine three envelopes. Until you look at the table and see only two!