Simple proof

You need to understand the concept of mathematical expectation. If you don't have that concept, I'd recommend the book *The Mathematics of Gambling* by Dr. Edward O. Thorp, or any text on elementary probability.

One fundamental, useful thing about expectation is that it's additive. In other words, if you have a list of mutually exclusive events, that together comprise the overall event you're interested in, you can multiply them by their individual probabilities, and then add to get the overall expectation. The "law of large numbers" tells us that the expectation is what you can expect to happen "in the long run."

For example: If you have three events, A, B, and C, which are mutually exclusive, and if the occurence of any of A, B, or C means that some improtant event E has occurred, then:

P(A) x Exp(A) + P(B) x Exp(B) + P(C) x Exp(C) = Exp(E), where P(whatever) is the probability that whatever occurs, and Exp(whatever) is the expectation if whatever occurs.

The concept is not limited to three events, it could be 1003 events or 1,000,003. Doesn't matter. Exp(whatever) can be any positive or negative number. Usually, for our purposes, it's a positive or negative dollar amount. Eg. if I bet $5 on the flip of a fair coin, and lose the whole $5 if tails comes up, but win only $3.65 if heads comes up, my overall expectation is 0.5 x (+$3.65) + 0.5 x (=$5) = -$0.675. Heads and tails are "mutually exclusive" -- they can't both happen. But if one or the other happens, we've completed the event in question -- a coin flip.

P(whatever) is always a number between 0 and 1, because P(whatever) = 0 means whatever is impossible, and P(whatever) = 1 means whatever is certain to occur.

What progressionists are saying is that they have a way to divide up an event, where each individual expectation is negative, yet the overall expectation adds up to something positive. Eg. Suppose Exp(A), Exp(B) and Exp(C) were **all negative**, they are saying that, nevertheless:

P(A) x Exp(A) + P(B) x Exp(B) + P(C) x Exp(C) = **something positive**

Remember that P(A), P(B) and P(C) are all between zero and one. Can't happen.

ETF