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.
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