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Comment by HelloNurse | Hacker News Reader

In the long time limit, there are about the same number of heads and tails, and since all changes are multiplicative, the coin tosses can be permuted. We can thus divide the game into two sets of coin tosses: excess heads or tails that represent a negligible amount of good or bad luck, and many HT pairs, each of which shrinks EV by 0.9.

Put another way, for each HH pair you should expect a TT pair, and a HHxTT or TTxHH sequence is worth the same as HTxHT, losing the usual 10% every two tosses. 2.250.36=0.81=0.90.9.

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Thanks, putting it this way does help me understand a bit better. So a large number of repeated fair tosses can be broken down into a sequence of win-loss pairs, which are always =0.9.

The thing that still confuses me is, why the heck is the EV 1.05? It seems to be expressing something true - if you were to split your money into a thousand piles and "play" each individually, you make money overall.

The gamble is set up in such a way that there is a fairly large chance that you will lose money, coupled with a tiny chance of an incredibly large gain.

The article, just like the poster above you, characterizes a series of bets into win/loss pairs that add up to a 0.9 return per pair. There are lots of sequences that can be characterized this way. However there will be a few sequences that contain many, many heads and win a lot of money. There are of course also a few sequences with many tails, but their loss cannot decrease below zero so it is contained.

So it's a little bit like a lottery ticket, where the positive gains are extremely concentrated into a very small lucky group. The more rounds of the gamble you play, the smaller the lucky winning group gets, and the larger their wealth.

> It seems to be expressing something true - if you were to split your money into a thousand piles and "play" each individually, you make money overall.

This is exactly the catch of the experiment. It's proposing you cannot split your money in independent experiments. You need to pick a history and stick with it. That's what ergodicity is about.

Hence, no one owns the average (EV) money of all possible outcomes and it's a pointless metric, even though it doesn't say something that's mathematically false.

The * characters got stripped from your message and interpreted as italics markers. What you probably wanted to write:

2.25 * 0.36 = 0.81 = 0.9 * 0.9.

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