There is no long term in this setup. There is a single offering. If we want to consider a long series of such activities, it needs to be set up precisely, and that has not been done. A number of crucial conditions will need to be specified, such as whether X is always the same and if not, how it varies from one play to the next. The best strategy will vary according to how those conditions are set up.If you could prove that always switching is the best strategy over the long term, doesn't that amount to proving that you are more likely to have chosen the smaller envelope? Why doesn't that bother you? — Srap Tasmaner
we use conditional probabilities to calculate the gain G from switching as follows, where the envelope we looked in contained £10:
E[G] = 0.5 * E[G | X= 5] + 0.5 * E[G | X= 10] ] . . . . . . . . . . . (0)
= 0.5 * E[-X | X= 5] + 0.5 * E[+X | X= 10] . . . . . . . . . . . (1)
= 0.5 * E[-5 | X= 5] + 0.5 * E[+10 | X= 10] . . . . . . . . . . . (2)
= 0.5 * (-5) + 0.5 * (+10) . . . . . . . . . . . (3)
= £2.50 . . . . . . . . . . . . (4) — andrewk
If we want to consider a long series of such activities, it needs to be set up precisely, and that has not been done. A number of crucial conditions will need to be specified, such as whether X is always the same and if not, how it varies from one play to the next. The best strategy will vary according to how those conditions are set up. — andrewk
If we want to consider a long series of such activities, it needs to be set up precisely, and that has not been done. A number of crucial conditions will need to be specified, such as whether X is always the same and if not, how it varies from one play to the next. The best strategy will vary according to how those conditions are set up. — andrewk
A number of crucial conditions will need to be specified, such as whether X is always the same and if not, how it varies from one play to the next. The best strategy will vary according to how those conditions are set up. — andrewk
Good, so you are challenging line 0 of my proof.Your error, BTW, is having the two situations are that X = 5 or that X = 10 — Snakes Alive
You have written a program with repeated plays that shows it is best to switch, and Srap appears to have written one that is only slightly different and shows that it makes no difference. — andrewk
Not if it is in two different games, which is what is happening here.If I have the larger envelope, then the other has 5 and the average value of an envelope is 7.5.
If I have the smaller, then the other has 20 and the average value of an envelope is 15.
(It is prima facie absurd that the average value of an envelope changes depending on whether you have the larger or the smaller of the two.) — Srap Tasmaner
the two possibilities that are open to me — andrewk
I mean that I know my envelope has 10 and that the other envelope has either 5 or 20.What does that mean? You know one of them must be in the distribution of X, but you don't know which — Srap Tasmaner
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