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Comment by throwaway1967 | Hacker News Reader

throwaway1967

This Platonic idea that just because it works on theory, therefore reality works just the same, is an old misunderstanding.

But before I criticize these "attempts" at solving this problem by these deft mathematicians, I'd like to see a single video example of 10 coin tosses all coming up either heads or tails.

After we can all see that this can happen, then we can start worrying about how much money a casino in the real world would charge for such game being played in the real world, by real players, with real coins.

It is amazing that out of all mathematicians listed on Wikipedia that attempted this, only one considered actually sampling (supposedly simulated coin tosses).

This can be easily simulated, but they'd rather stay within the comfy confines of calculation, so that presumably they can publish more papers.

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http://www.youtube.com/watch?v=rwvIGNXY21Y

Not that hard to do. Just a matter of perseverance and a bit of luck. Yes, it could be faked, but I trust him.

Similarly, here's another fellow rolling a Yahtzee. https://www.youtube.com/watch?v=fiTwar7mFws

Tossing the coins can be useful to help our intuition, but we have a pretty good understanding of how they work, to put it mildly. I think it's fair to assume that a mathematician isn't going to change their mind after a quick experiment.

If you asked for, say, 40 heads in a row, I could see it. But 10? That's merely a one-in-a-thousand occurrence. Or one-in-five-hundred if you accept either heads or tails.

If only he/she actually bothered to throw coins before writing that comment...

https://youtube.com/watch?v=rwvIGNXY21Y

And there are several more.

A coin toss isn't exactly 50% for both sides in a real life scenario, hence the difficulty to reproduce the experiment exactly like its theoretical description. A sequence of 10 coin tosses resulting in heads/tails is not impossible even if it is hard to reproduce, it simply has an absurdly low probability of ocurring.

This focus on the theoretical side of things, the 'comfy confines of calculation', isn't a bad thing, the paradox is still an interesting thought experiment.

The paradox does not depend on the fairness of the coin.

A real coin toss is, say, at least 25% probability for heads and at least 25% probability for tails. Therefore you can multiply the winnings by 4 and not by 2, which will give you a divergent series even with a real, "unfair" coin.

So then, you have an answer for how much a person should be willing to pay to play the game, based on your simulation approach?

I don't get your idea with sampling. It will work in practice. To see that yourself: write a program that repeats this experiment again and again, compute the averages of the winnings, and you will see that they cannot be bounded by any constant. Or use a real coin and a piece of paper, though this will require a lot of patience.

While it cannot be bound by any constant, it is described by a function over number of tosses. When you introduce an entry price, you will suddenly get negative values in the beginning, and can reason again against expected value over time much better.

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