Wednesday, April 13, 2011

Twice the Fun at Half the Price

Someone shows you two boxes and he tells you that one of these boxes contains two times as much as the other one, but he does not tell you which one this is. He lets you choose one of these boxes, and opens it. It turns out to be filled with $10. Now he gives you the opportunity to choose the other box instead of the current one (and give back the $10 from the first box), because the second box could contain twice as much (i.e. $20).

The Question: Should you choose the second box, or should you stick to your first choice to maximize the expected amount of money?


  1. Choose the first, with the ten dollar bill. The other box does not have a twenty dollar bill, because if it did then that would imply the existence of twice that amount (i.e. a forty dollar bill) would be in the other box (had one chosen the second box with the possible twenty dollar bill in it first). There are no forty dollar bills.


  2. Anonymous (5:45PM) is assuming that the choices are $10 and $20. It could just as well be $5 and $10 or it could be $10 and &20. Either way, stay with the first pick. I suspect, were I of such knowledge, the odds are 3 to 1 in favor of the first pick versus a second pick.

  3. Turns out there is no way to maximize your outcome here. You still have a 50/50 chance that you picked the maximum return on your first pick.

    So, flip a coin if you want, but as Hank points out, the other box could have either $5 or $20 in it. They are both just as likely.

  4. this was an old question right? i mean approx 5 years ago.. and still confuses me.. :p


  6. I hate it when I re-run old questions. But after five years, it's hard to remember them all.

    Basically, when you picked the first time, you had a 50-50 shot at picking the larger bill. When choosing to switch or not, you haven't added anything new to your information. Either box could still hold the larger amount, so you still have a 50-50 shot at already holding the larger amount.

  7. what is there beyond universe ?

  8. I'm not very good with statistics, but is there a way to do expected value analysis of this? 0.5 * 5 + 0.5 * 20 = $12.5, Meaning that if you kept playing this scenario over and over again you would be richer by not going for the $10?

    If you see $20 instead, the other box would have $10 or $40, with an EV of $25...?

  9. Your potential gain with the second box is bigger than your potential loss, so you should choose the second box.

    Think of it this way: if the ratio between the boxes was not two, but a million, you'd have a 50% chance of either getting essentially nothing or gaining a million.

    I know what I'd choose.

  10. This is actually pretty similar to the Monty Hall problem.

    You should switch doors!


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