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Topic: A good probability puzzle but what is the right wording?
Replies: 10   Last Post: Feb 24, 2013 12:19 PM

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Posts: 12,067
Registered: 7/15/05
Re: A good probability puzzle but what is the right wording?
Posted: Feb 4, 2013 8:15 AM
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quasi wrote:
>Paul wrote:
>>The following puzzle is copied and pasted from the internet.
>> Alice secretly picks two different real numbers by an unknown
>> process and puts them in two (abstract) envelopes. Bob
>> chooses one of the two envelopes randomly (with a fair coin
>> toss), and shows you the number in that envelope. You must
>> now guess whether the number in the other, closed envelope
>> is larger or smaller than the one you?ve seen. Is there a
>> strategy which gives you a better than 50% chance of guessing
>> correctly, no matter what procedure Alice used to pick her
>> numbers?

>Let R denote the set of real numbers and let (0,1) denote
>the open interval from 0 to 1.
>Let f : R -> (0,1) be a strictly decreasing function.
>Use the following strategy:
>If the initially exposed value is t, "switch" with probability
>f(t) and "stay" with probability 1 - f(t).
>Suppose Alice chooses the pair x,y with x < y (by whatever
>process, it doesn't matter). After Alice choose that pair,
>then, by following the strategy I specified above, the
>probability of guessing the highest card is exactly
> (1/2)*f(x) + (1/2)*(1 - f(y))
>which simplifies to
> 1/2 + f(x) - f(y)

I meant:

which simplifies to

1/2 + (1/2)*(f(x) - f(y))

>and that exceeds 1/2 since f is strictly decreasing.
>Of course, it's not the case that probability of guessing
>correctly is more than c for any fixed c > 1/2, but the
>problem didn't require that.


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