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Probability problem

DoublemeDoubleme Member Posts: 2,150
in The Rail
So in this scenario you are on a game show and about to win a prize your prize is in an envelope which has a cheque in it for an unknown amount. All envelopes have at least £1000 in as a price could be much more though. Now you are informed that there is a 50% chance that the other envelope is worth twice the value of whatever is in your envelope. there is also a 50% chance that the other envelope is worth half what your envelope is.

suppose then that it will cost you £1 to switch envelope should you do so? since we do not know the value of our envelope we can say our envelope is worth X. the alternate envelope has a 50% chance of been worth 2X and a 50% chance in been worth 1/2(X) .

So we can work out the expected value of our choices if we stick with our original envelope then our expected value is just X.
If we switch then our expected value is worth 0.5*(2X)+0.5*(1/2)X=1.25X

now since we know that X is worth at least £1000 the value of switching for only £1 seems a no brainer right. we are spending £1 to gain at least £250 of EV how can we turn that down?

so you switch now the game show host offers you the choice of do you want to switch back for £1? well we know the other envelope is worth either twice the value of ours or half the value of ours. now then we are in the same scenario our envelope is worth an unknown amount which we can call X and the other envelope will be worth 1.25X so should we switch? well it is the same problem as before so the answer is yes.

So we keep switching indefinitely and go broke.

What is going wrong here?
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Comments

  • goldongoldon Member Posts: 9,061
    And they think I'm the
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  • dragon1964dragon1964 Member Posts: 3,054
    Wrong.
    Originally 50% chance of winning £1000, 50% chance of losing £500.
    Second offer.
    50% chance of winning £500, 50% chance of losing £1000.
    Don't swap again.

    I think.🤔
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  • DoublemeDoubleme Member Posts: 2,150
    dragon1964 said:

    Wrong.
    Originally 50% chance of winning £1000, 50% chance of losing £500.
    Second offer.
    50% chance of winning £500, 50% chance of losing £1000.
    Don't swap again.

    I think.🤔

    first you read it wrong it said at least £1000 for all envelopes. so the other envelope cannot be worth £500 either one of the envelopes could be worth £1000 £2000 £4000 etc or even £250,000 etc etc the issue is one of the envelopes is worth twice as much as the other one but you dont know which way round this is.

    what I have posted is actually a famous probability question, though the wording of £1000 at least is my own. the generic problem is that you have to pay a tiny amount relative to the prizes in order to switch an amount which appears utterly immaterial.

    the probability argument from the point of expected value would indicate that you should switch every single time but obviously something is wrong here.

    It is actually a famous paradox.
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  • dragon1964dragon1964 Member Posts: 3,054
    edited March 2022
    Yes, sorry, I read it wrong.

    Ignore me.
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  • EssexphilEssexphil Member Posts: 8,782
    edited March 2022
    Doubleme said:

    So in this scenario you are on a game show and about to win a prize your prize is in an envelope which has a cheque in it for an unknown amount. All envelopes have at least £1000 in as a price could be much more though. Now you are informed that there is a 50% chance that the other envelope is worth twice the value of whatever is in your envelope. there is also a 50% chance that the other envelope is worth half what your envelope is.

    suppose then that it will cost you £1 to switch envelope should you do so? since we do not know the value of our envelope we can say our envelope is worth X. the alternate envelope has a 50% chance of been worth 2X and a 50% chance in been worth 1/2(X) .

    So we can work out the expected value of our choices if we stick with our original envelope then our expected value is just X.
    If we switch then our expected value is worth 0.5*(2X)+0.5*(1/2)X=1.25X

    now since we know that X is worth at least £1000 the value of switching for only £1 seems a no brainer right. we are spending £1 to gain at least £250 of EV how can we turn that down?

    so you switch now the game show host offers you the choice of do you want to switch back for £1? well we know the other envelope is worth either twice the value of ours or half the value of ours. now then we are in the same scenario our envelope is worth an unknown amount which we can call X and the other envelope will be worth 1.25X so should we switch? well it is the same problem as before so the answer is yes.

    So we keep switching indefinitely and go broke.

    What is going wrong here?

    You are using outdated parameters for an updated scenario.

    Every envelope is worth double or half. But no envelope is worth 4x any other envelope. So the 2nd swap cannot contain double wins or double losses.

    Therefore it necessarily follows that you will lose the theoretical gain you have previously made via the 1st swap. As in all cases you will be back to your original amount.

    So-£2k-4k-2k
    Or-2k-1k-2k
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