Science Forums 2

CraigD · 06-25-2009

Quote:

Originally Posted by lawcat

The best thing is not to switch...

This is correct, but misses the point of the paradox – see the wikipedia article “two envelopes problem” Erasmus linked to for a decent encyclopedic description.

In short, the puzzle is to explain the paradox of how the ordinary, elementary, and usually trustworthy mathematical statistical technique of expected value seems to go wrong to produce a paradox in which it appears to instruct you that you should always change your selection. It’s a whodunit mystery, where the perpertrator of the crime is some step in reasoning or assumption.

If one knows the exact numeric outcomes $V_1,\,V_2,\,\dots\,,V_n$ (AKA payoff) and probabilities $p_1,\,p_2,\,\dots\,,p_n$ of each occurring for an event, the expected value of it is
$V_{\mbox{expect}} = V_1P_1+V_2P_2\,\dots\,+V_nP_n$ .
For example, the expected value of a fair 6-sided die roll is
$1\frac16+2\frac16+3\frac16+4\frac16+5\frac16+6\frac16 = \frac{21}6 = 3.5$
which can be confirmed by noting that, after many die roles, the average roll approaches 3.5.

The box/envelope question seems to lend itself to this same approach (if it doesn’t, the philosophical underpinnings of mathematical realism are in serious peril!

) – but if stated in the usual way (see the wikipedia article), gives:
$V_{\mbox{expect}} = 3x\frac12+\frac{x}3\frac12 = \frac{5x}3$
where $x$ is the amount in the first box you pick, and $\frac{10x}6$ is the expected value – what you should, on average, get, if you change you choice to the other box. So you should always change you choice, and change it as many times as you’re allowed, which is wrong, and just plain silly.

(Note that the wikipedia article’s version states that one envelope has 2 times the payoff of the other, giving $V_{\mbox{expect}} = \frac{5x}4$ , while post #1 has it as 3 times, but this doesn’t change the essence of the paradox)

To appreciate the paradox, the reader should try to understand what’s wrong here (trust me and wikipedia – the philosophical underpinnings of mathematical realism are not in jeopardy here

). Appreciated well, this paradox build intuitive and formalizable comprehension of what a value is in mathematics, and how they correspond to commonplace events like this problem’s.

I’ve what I think’s a pretty good, formally sound yet intuitively strong, explanation of this paradox, but don’t want to spoil it for people new to the paradox, so will wait to see what others come up with. Enjoy, all, and thanks, Phillip, for starting this thread

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