The paper attached describes a scenario where the player A chooses a number between -infinite to infinite. The simulation uses ranges -1000000 to 1000000, it's not a surprise that the strategy works in this case.
I think if the ranges were truly -infinite to infinite, this strategy would fall apart. No matter where you set C, there are infinite values above and below it.
The game only makes sense if you attach limits to the numbers the other person is allowed to pick, and then the only "split" that works is in the middle of the range. At which point the game is sort-of "obviously true".
If the distribution of A and B is also a proper distribution the probability that C is between them is positive. But then the distribution of A and B has a middle point such that 50% of the numbers are above and 50% below and the problem is trivial.
But for any two finite numbers, A and B, the chance that C will fall between them if selected from an infinite range is 1/infinity (or 0).
I think you're assuming we pick C uniformly at random from an infinite range, but this is not possible. In general, the claim is not true: consider picking C from a sample of a standard normal, if A = -B = 2 then with >95% C will lie between A and B.
It's equivalent to this much simpler game: "I am going to give you a positive real-number amount of dollars." No matter what I do, you will win money playing this game, guaranteed! Now, how much would you pay to play this game? $0, because for any amount you'd pay to play, I could arbitrarily choose to give you less in winnings.
Like most problems involving infinity, it's unintuitive/paradoxical because it pretends to model a physically plausible scenario but actually doesn't.
The probability that a random number uniformly drawn from the real line is between 42 and 43 is zero in the same way that “the probability that a random real number selected uniformly from the interval [0 1] is pi/4” is zero.
Integral of 1/2^n ... got that no problem.
I took AB Calculus in high school, three semesters of calculus in college, and a bunch of calculus heavy linear algebra. I’m certain that I knew how to integrate that thing at some point... it’s pretty sad that now I can only stare at it blankly and at best lean heavily on Wikipedia to find an answer.
https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%...
No. It doesn't even require Player A to pick them at random. The strategy for Player B still works over 50% of the time.