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# The two envelopes problem

factor-fiction | 15:15 Thu 10th Nov 2011 | Riddles
The Deal or No Deal thread reminded me of two tricky probability/choice problems. One is the Monty Hall goat and cars problem which has been covered on here several times before so I'll leave that one.The other is the Two Envelopes problem:

Suppose you are in a Game Show. The host has 2 identical looking envelopes The host then tells you that one envelope contains twice as much as the other but neither of you knows which is which.

He asks you to choose one envelope and says you can open it and he will then offer you the chance to 'swap' or 'stick'.

You pick an envelope and open it to find it contains £2400. You think
"Mmm. The other envelope contains either £4800 or £1200. Should I swap or stick?"

What would you do?

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I've just read this all the way through, now I need to go and lie down.....
http://en.wikipedia.o...Two_envelopes_problem
15:24 Thu 10th Nov 2011
Id stick, and hope that I wasnt shown what the other envelope contained :)
I'd certainly stick because doing so would mean that there was no possibility that I could leave with less money than I had at that moment.
i'd swop
Question Author
Interesting answers so far.

Mark- if the rules were different and you were told one contained ten times as much as the other would you still stick?

Sammo- Can you tell us why you would swap?
If ten times more also meant ten times less, then I would certainly stick.
So I risk £1200 as a stake, and on a 50:50 chance could win £2400
I'd swop.
I've just read this all the way through, now I need to go and lie down.....
http://en.wikipedia.o...Two_envelopes_problem
I was looking at this problem quite recently and somewhere in my in-tray is some proof (by a Swedish woman???) that you should always swap - or was that something else?
Question Author
That may well have been the more famous Monty Hall goats and cars problem, Prudie.
I've just read sidkid's link and I feel a headache coming on.
"and says you can open it and he will then offer you the chance"
Are you sure you were supposed to look in it first ?
I'd stick... bird in the hand and all that
Question Author
There are two variations of the problem, OG. Do you think it makes a difference whether you see the amount or not?
In your example I would take the £2400.

Like Mark I would stick if you changed the multiplier to 10 times.

If the amounts were greater, say £24,000 in the first envelope, and either £48,000 or £12,000 I would swap. I think pay off would be worth the risk for the potential higher pay-out.
You're right factor, sorry. I've just found it in my tray!. It is the Monty Hall goats and cars but as that was also set for a game show scenario my memory was selective.
Ed...first scene of AbEd the musical is on apotheosis
It's a variation of the old joke:

60 year old man to good looking 25 year old woman: "Would you go to bed with me for £50000"

"Yes, I would."

"Would you go to bed with me for £50?"

"What the hell do you think I am, a ***?"

"We have established the fact, we are just negotiating the price."
Not as such it's just that I checked out sidkid's link and it stated the poser differently and I wondered if you had it correct.

However I may be missing something but it seems to me that wiki page just gets too complex for it's own good. It starts to state nonsense regarding it being beneficial to continue to swap envelopes indefinitely. But of course that fails when you realise that in that scenario each step isn't separate. Inevitably if you gain on the first swop you will lose on the second and are back as you started.

I still say swop. Maybe one can run a spreadsheet and prove it works for separate decisions over a large number of choices.
Question Author
Ho O_G. I suppose the question of seeing the amount or not seeing it could become relevant if you knew there was an lower and upper limit on the values- for example if it was known that the amounts ranged from £1 to £1048576 and you picked one of the extremes you'd be able to work out for sure what the other one was.
so it's like OG says, a double-or-nothing bet. I'd risk losing £1 to gain £2 but I think with £2400 I'd stick.

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