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[Not-so-daily puzzle] Two envelope puzzle
900 sats \ 11 comments \ @south_korea_ln 25 Dec science
A classic puzzle for Christmas.
You are presented with two sealed envelopes. Inside each envelope is some amount of money, and you know the following:
  • One envelope contains exactly twice as much money as the other.
  • You are allowed to choose one envelope and keep the money inside.
After you select an envelope, the game host offers you a deal: you can switch to the other envelope if you want.
You reason as follows:
Let the amount in the envelope you chose be .
If you switch, the other envelope could either:
  • Contain (if our current envelope contains the smaller amount), or
  • Contain (if your current envelope contains the larger amount).
On average, switching seems to give you:
This is greater than the you currently hold, so your reasoning seems to indicate you should always change!
What's the mistake in this reasoning? As this is a famous problem, please don't look up the solution online and only answer if you haven't heard of this problem before~~
Previous iteration: #817694 (solution in #817789)
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300 sats \ 2 replies \ @Undisciplined 25 Dec
The problem is that one of those "x"'s is twice as large as the other.
You have to use something like x and 2x for the smaller and larger quantities, respectively. That gives you an expected value of 1.5x from switching, which is equal to the expected value of what you have.
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33 sats \ 1 reply \ @south_korea_ln OP 25 Dec
That's right. Not a big challenge for you to spot this error, I'm sure.
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0 sats \ 0 replies \ @Undisciplined 25 Dec
Opportunity cost, baby!
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145 sats \ 5 replies \ @SwapMarket 25 Dec
Answering this correctly many years ago got me a job on Wall St. You don't have to look inside since the expected value after switch is always 1.25 higher. So if you accept this logic you will keep switching unsealed envelops forever. The answer is not to switch because there is no new information gained when looking inside the envelope. The amount you see does not help you decide.
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11 sats \ 2 replies \ @SimpleStacker 18h
I thought the way to get a job on Wall Street was to solve a Rubiks Cube :D
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1 sat \ 1 reply \ @Bell_curve 11h
#818230
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0 sats \ 0 replies \ @nitter 11h bot
https://xcancel.com/DudespostingWs/status/1870128444849180977
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0 sats \ 1 reply \ @south_korea_ln OP 23h
Wow, quite fascinating this is what got you a job inside Wall St.
You are right on this:
The answer is not to switch because there is no new information gained when looking inside the envelope.
but it makes me wonder how many Stackers know about the Monty Hall problem. I never posted it as I assumed everyone has heard of it at some point. There, the new information being gained condition is harder to assess...
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145 sats \ 0 replies \ @SimpleStacker 18h
Fun fact, your puzzle today inspired me to teach my kids the Monty Hall problem. At first they were like, "What difference does it make?" But they seemed to understand when explained. Hope they remember it
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0 sats \ 1 reply \ @ama 25 Dec
I think it's equivalent of some when price goes up 100% it only needs to drop by 50% to get to the same value. That is, multiply by 2 and divide by 2 "don't yield the same result". If that makes sense. :-)
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0 sats \ 0 replies \ @south_korea_ln OP 23h
I think I see what you are hinting at. But I don't think it's related. Here, it is really just because one should not use the same variable for the two cases, as they are different. See #823807.
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