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# Whether to switch

22 replies to this topic

### #1 bonanova

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Posted 02 May 2013 - 07:45 AM

Is there no end to circumstances that ask whether we should switch?

This one might take some thought.

Your rich uncle places a sum of money into an envelope, twice that amount into a second envelope, places them both into a black bag and invites you to draw one out of the bag. You open the envelope that you draw and find \$1000. You now know the envelope in the bag contains either \$500 or \$2000. Your uncle gives you the option to take the other envelope in exchange for the one you drew.

Calculate the expected gain that comes from switching.

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Vidi vici veni.

### #2 bubbled

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Posted 02 May 2013 - 12:32 PM

Assuming that your Uncle has \$3000, so he could stuff one envelope with 1000 and the other with 2000:

Spoiler for My answer

The real issue with all of these puzzles are the definitions. I don't think it's worth anything to switch, unless you have a clearly defined range of possible amounts in the envelopes. The possible range of values must have an upper limit, unless we are to accept a possible infinite envelope (I'm not). Let's say the range of possible values is 1-100 inclusive. And all possible parings are equally likely. Then the actual best strategy is if you open an envelope with 50 or less in it, you switch, otherwise you stand pat.

If you're not given a clear range, then a perfect strategy is not possible to construct. For instance, if the range was 1-1,000,000, but you didn't know it, all the value you might gain by switching, will be lost when you draw an envelope higher than 500,000 and switch into an envelope that must be smaller.

Edited by bubbled, 02 May 2013 - 12:32 PM.

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### #3 vigmeister

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Posted 02 May 2013 - 05:35 PM

Spoiler for If you knew before

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### #4 k-man

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Posted 02 May 2013 - 09:02 PM

Spoiler for my take

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### #5 bonanova

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Posted 03 May 2013 - 09:09 AM

Spoiler for my take

Can you gain as much by switching back? Using the same reasoning?

There is a paradox: A gain for switching can be anticipated.

Yet, there is a preferred envelope, and if we initially chose it we should not switch.

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Vidi vici veni.

### #6 bubbled

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Posted 03 May 2013 - 10:42 AM

I would say no. The switching in and of itself is not what causes the gain in expected value. Assuming that the chances of 500 or 2000 (in your example) are equally likely in the other envelope, then the gain in EV, comes from calculating that value and determining it is higher then 1000, and then switching. Even though you haven't looked into the other envelope, its value can be calculated and switching back would lose the EV gain you made by switching the first time.

If you were to know the exact distribution (possible values and frequencies of those values) of all envelopes, then before you look into any envelope, you can assign it an expected value of an unknown random envelope. At that time, switching would not gain any EV, because absent any additional information, the other envelope has the exact same EV. Once you look into the first envelope, by knowing the overall distribution of all envelopes, then adding in the new and valuable information you have gleaned by reveling the contents of the first envelope, you can then make a proper choice as to whether or not to switch.

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### #7 plasmid

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Posted 04 May 2013 - 03:35 AM

Let's see who can tell us what's wrong with this line of reasoning.

First, consider the case where the amount of money in the envelope could be any real number, not restricted to integers or whole cents.

Spoiler for

Next, consider the case where the amount of money in the envelope must be some integer value of dollars and/or cents. Let us also assume for the sake of the problem that there is an equal probability that the smaller envelope contains any possible amount of money, odd or even.

Spoiler for

I believe this proves all possibilities: one where you're more likely to have the smaller amount of money, one where you're more likely to have the larger amount of money, and one where they are equally likely.

Edited by plasmid, 05 May 2013 - 04:52 PM.

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### #8 bhramarraj

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Posted 06 May 2013 - 12:52 PM

Spoiler for

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### #9 k-man

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Posted 06 May 2013 - 08:04 PM

Spoiler for my take

Can you gain as much by switching back? Using the same reasoning?

There is a paradox: A gain for switching can be anticipated.

Yet, there is a preferred envelope, and if we initially chose it we should not switch.

Not sure what you mean by "switching back". The 50/50 comes from randomly picking one of 2 envelopes from which we know one has double the money of the other. I don't see a paradox.

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### #10 Brainwiz

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Posted 07 May 2013 - 02:15 PM

I think not worth it to switch - there's no gain.

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