I don’t think this explanation is very good. We could use a distribution where the expectation is defined and be back to square one. For example you can just tweak his distribution so that the ratio of envelopes over the ratio of successive probabilities is < 1 (he has 10:2 which is >1).

Yeah the Wikipedia explanation is terrible! Someone rewrite that.

that's an excellent treatment. I'm curious: this video picks a particular example of a distribution of the amounts and shows that the expected profit from switching is not defined. Can one somehow prove that for all possible distributions, either the expected profit from switching is 0 or it is undefined?

the point of the video is that given a distribution, the total expected profit cannot be defined, because the infinite sum of the probabilities of each case don't have an order, and also adds up to positive infinity and negative infinity.

it doesn't really matter what the distribution of the amounts are, as long as there's an infinite number of possibilities in the distribution (ie., it's not a finite amount of possible envelopes).

That’s not true for all distributions.

For example if instead of halving the probability of each value as he does in the video we take 1/20 then we get the expected value as the sum of 9/20^2 - 9/20^2 + 90/20^3 - 9/20^3 + … Here the series is absolutely convergent since the positive terms sum to 9/10 and the negative terms sum to 9/10.

> halving the probability of each value as he does in the video we take 1/20

but then the probabilities don't all add up to 1.

> as long as there's an infinite number of possibilities in the distribution

how would one prove that claim? it's not obvious to me. it seems like the specific conditions under which the series of terms in the total expectation has a well defined sum is important, but I didn't fully grasp those conditions