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Comment by aegiso | Hacker News Reader

The way I see it the flaw is in the first sentence of the "example":

> Assume the amount in my selected envelope is $20.

You can't just pull that assumption out of your ass. You can only assume what the problem states, which is that the values in the envelopes are X and 2X and you had a 50% chance of choosing either.

If you run the math without adding assumptions, it works out that swapping makes no difference, statistically.

Though it is a very clever trick :).

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Taking out the specific dollar amount doesn't change the math at all. If X designates the amount in the envelope I've selected, than 50% chance the other envelope contains .5 * X and a 50% chance it contains 2 * X, so the expected value of the other envelope is .5 * .5 * X + .5 * 2 * X = 1.25 * X which is greater than X.

but you are implicitly turning the scenario into one with three values, 1/2X, X, and 2X, so something went wrong with what you're doing. There was only ever a universe of two values.

Sure, these are variables, so they correspond to different possibilities. X can be anything, and given X, the amount in the other envelope is one of two different possible values - 1/2X OR 2X.

So there are far more than three possible values, but only two actual values.

Exactly, that's why your model is broken.

> If you run the math without adding assumptions, it works out that swapping makes no difference, statistically.

I beg to differ :)

https://news.ycombinator.com/item?id=6387344

I think the key assumption is there is a 50% chance of getting double and a 50% chance of getting half in the swap scenario. Once you pick an envelope initially the chance disappears; you can't parlay the chance into the new context.

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