-----Original Message----- From: Stathis Papaioannou [mailto:[EMAIL PROTECTED] Sent: Friday, October 15, 2004 12:35 AM To: [EMAIL PROTECTED] Subject: RE: Observation selection effects

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Brent Meeker wrote: QUOTE- It's not wrong - I just don't think it addresses the paradox. To resolve the paradox you must explain why it is wrong to reason: I've opened one envelope and I see amount m. If I keep it my gain is m. If I switch my expected gain is 0.5*m/2 + 0.5*2m = 1.25m, therefore I should switch. -ENDQUOTE The reason this is wrong is that "m" is not a constant for the purposes of the probability calculation. I open one envelope and I see amount m. I know that one envelope contains x and the other 2x, but I don't know whether m=x or m=2x at this point. Pr(m=x)=0.5 and Pr(m=2x)=0.5. So my expected gain if I keep the envelope is 0.5*x + 0.5*2x = 1.5x. My expected gain if I switch is the same. The obvious response to the above is to point out that "m" IS a constant, and if I keep the first envelope my gain is exactly m, whatever "m" may be in terms of "x". This is true, but I need to know what m is in terms of x if I am to go on to calculate the probabilities in the situation where I switch: (a) If I happen to have opened the envelope first which makes m=2x (with Pr=0.5), then there is a 100% probability that the second envelope contains 0.5m=x and 0% probability that the second envelope contains 2m=4x - so that my expected gain is 0.5x (or 0.25m) from this half of the decision tree. (b) If I opened the envelope first which makes m=x (with Pr=0.5), then if I switch there is a 100% probability that the second envelope contains 2m=2x and 0% probability that it contains 0.5m=x - so that my expected gain is x (or m) from this half of the decision tree. Now, to get the final expected gain from switching, I add up the result from (a) and (b). If I add up the x's I get 0.5x + x = 1.5x, the same as if I had kept the first envelope. If I add up the m's, I get 0.25m + m = 1.25m, which seems to be greater than the m I would get if I kept the first envelope, as per your analysis above. Which should it be for comparison purposes, 1.5x or 1.25m? I think you can see that whereas x is a constant, m is one amount in (a) and a different amount in (b). You can't add up the m's in the two parts of the decision tree as if they were the same. Stathis Papaioannou _________________________________________________________________ In the market for a car? Buy, sell or browse at CarPoint: http://server-au.imrworldwide.com/cgi-bin/b?cg=link&ci=ninemsn&tu= http://carpoint.ninemsn.com.au?refid=hotmail_tagline