Comment by jsweojtj | Hacker News Reader

Comment by jsweojtj | Hacker News Reader

Parent Full thread

The ensemble average is the expected value, and the expected value is positive.

For a bet of $X, the expected value is: (1.5 * X) * 0.5 + (0.6 * X) * 0.5 => 1.05 * X. The ensemble average per round is positive (1.05) and over multiple rounds smoothly tends to infinity with the number of bets. (Definition here: https://en.wikipedia.org/wiki/Expected_value).

The time average for any specific person betting in this game is 0.95 * X (for the reasons you mention) and tends to zero with the number of bets.

So let's go through a few specifics of your comment:

> He actually picked bad numbers. That is a losing bet even on average.

The point of this article is that "on average" is trickier than people tend to assume. There are different ways of taking averages. If you do the expected value calculation and get a positive number, you might (as other comments have said explicitly) expect that a participant repeatedly engaging such a bet would have his wealth trend toward infinity. But, they are wrong (as shown in the article).

> This happens to not show after only 100 trials just because some tiny number of people get really lucky and draw up the ensemble average, but if you keep going, somewhere between 200 and 500 trials, the ensemble average pretty quickly drops below the starting average wealth and stays there, asymptotically approaching 0.

The ensemble average is positive and monotonically increases w/ the number of rounds of betting.

I'm getting too into the weeds below with these simulations, but I think it maybe speaks to this author's point. I can see you are clearly right analytically. With a positive expectation on each round, the expectation itself clearly trends to infinity.

But I can also clearly see that never actually happens, and I do think I can explain it. The measure of people with any positive expected return at all after a large enough number of trials is so small that they eventually drop out of any actual simulation just because nobody ever gets that lucky, even though theoretically it is possible. You eventually reach a point after a large enough of trials (apparently about 500) where if anybody at all was actually hitting the 400+ heads out of 500 trials requires to still be above water, enough of them would be so fabulously wealthy that they'd draw up the entire average. But the probability of these trials ever happening is so low that we can run simulations for thousands of years and never see it happen, so what we see instead is nothing but common cases. Everyone after 500 trials is overwhelmingly within +/- 50 of 250 heads and 250 tails, and if you're in that range, you're a loser.

It also may be the case that the way these pseudorandom number generators work makes it completely impossible to ever see 400 heads out of 500 trials no matter how many simulations you run since they aren't actually random, but even if they were, practically speaking, I wouldn't be the least bit surprised if an any trial ever run of 500 consecutive fair coin tosses, nobody for as long as humans have existed has ever hit 400+ heads. If you try to analytically compute the probability and store it in a floating point number, it just rounds to 0 because we can't store a probability that low.

Yes, the pool of participants in your simulation must be large enough to sample all of the relevant phase space to generate the analytic solution.

This becomes very large as the number of trials increases (likely approximately exponential)

Reply on news.ycombinator.com