Answer to Puzzle #23: Envelopes, Stick or Change

23. I have three envelopes, into one of them I put a £20 note. I lay the envelopes out on a table in front of me and allow you to pick one envelope. You hold but do not open this envelope. I then take one of the envelopes from the table, demonstrate to you that it was empty, screw it up and throw it away. The question is would you rather stick with the envelope you have selected or exchange it for the one on the table. Why? What would be the expected value to you of the exchange?

With this puzzle it's not so much explaining the answer that's difficult, as it is convincing you I'm right... This is an example of the Monty Hall problem.

picture of three envelopes Before reading the answer can I interest you in a clue?

The short answer is that it is in your advantage to exchange. But why?

Well initially there was a 1/3 chance that you were holding the envelope with the note in it and a 2/3 chance that the note was on the table.

This is still the case after one of the envelopes on the table has been removed, there is still a 1/3 chance that you have the note and a 2/3 chance of it being on the table.

If this is confusing then it may help to think that the questioner knows which envelope the £20 note is in, though in practise it doesn't actually matter. The questioner would always be able to demonstrate that the note was not in one of the envelopes on the table regardless of where the note was, so the fact that he was able to do this changes nothing.

Consider a different example....

Say there are a 1000 envelopes on the table, 1 with a note inside. You pick 1 envelope, the chance that this has the note in it is clearly 1/1000, where as the chance that it is still on the table is 999/1000. Odds are its on the table. Now the questioner could, assuming he can remember where the note is demonstrate to you that the note is not in 998 of the envelopes on the table. In this case nothing would have happened to change the fact that there is only a 1/1000 chance of you having the note.

That is why you exchange.

What is the value of the exchange?

Simply before the exchange you have 1/3 of £20 and afterwards you will have 2/3 of £20, ie the advantage to you is about £6.66.

They both got this problem largely correct. However, they both made the same mistake... They miscalculated the value of the exchange, 'forgetting' the the position you're in also has a value.

If you're curious what Bard made of this puzzle...

If you're curious what ChatGPT made of this puzzle...

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