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Comment by aynsof | Hacker News Reader

Here's an experiment: get a million people to guess the results of a thousand coin-tosses. By the end, a miraculous few will have a 100% record and will be hailed as geniuses. Books will be written about this incredible group. These books will find the traits that, by chance, the group members happen to share with each other. ("My god, they all have brown hair!" "Yes, but they come from eclectic backgrounds - housewives, factory workers, math professors." "How fascinating!") But what is the probability any one member of that group correctly guessing the result of the next coin toss?

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you get one person to guess the result of 1000 coin tosses that's a 1/(2^1000) probability of one person being right. Get a million people to guess it it's (10^6)/(2^1000) probability of one person being right.

Lets put it in perspective; the chance of one person out of a million getting a 100% streak is 9.332 × 10^-296 %. That's significantly less than 0.000000000000000000000000001%.

If there are people who have a 100% record of being correct on every guess, the probability of them being geniuses is way higher than the probability of them being lucky.

That being said, I get your point. We don't have enough metrics from the research to determine whether or not these super forecasters are just lucky or geniuses.

It turns out not to matter (since the numbers are so small) but your math implies that 10^305 people guessing would give a probability of about 9,300 that at least one of them is right on all 1000 coin flips. Since probabilities can't be larger than 1, that's a bit of a problem.

For anyone interested, the full set of steps (that produces a numerically identical result):

    Prob[1 or more in 1,000,000 right]
      = 1 - Prob[all 1,000,000 wrong]
      = 1 - Prob[person 1 is wrong AND person 2 wrong AND ... person 1,000,000 wrong]
      = 1 - Prob[person 1 is wrong]^1,000,000
      = 1 - (1 - 0.5^1000)^1,000,000
      = 1 - exp(1,000,000 * log(1 - 0.5^1000))
      = 1 - exp(1,000,000 * log1p(-0.5^1000))
      ≈ 1 - exp(1,000,000 * -9.33 × 10^-302)
      = 1 - exp(-9.33 × 10^-296)
      = -expm1(-9.33 × 10^-296)
      = 9.33 × 10^-296

log1p(x) = log(1 + x) but is more accurate when x is near zero.

expm1(x) = exp(x) - 1 but again is more accurate when x is near zero.

Both are necessary here to get a result other than "0".

It's perfectly reasonable to approximate a+b=a+b(1-a) when combining rare events without announcing it to everybody. Likewise with n repetitions of that.

Fair enough, but it's also reasonable to approximate 9.33 × 10^-296 as 0. I felt like doing the calculations :)

You should read Tetlock's other work about foxes and hedgehogs in Expert Political Judgment. It presents a theory of why some people are indeed better forecasters than others. It's not lucky or genius, it's more about people who are able to consider several different data points, consider historical precedent, and make a relatively unbiased judgment about what's going to happen. These are the same types of people he sought out for the Good Judgment project, and its these same types of people who were considered Superforecasters making hundreds of forecasts over the course of a year about geopolitical issues.

You're right - that was a terrible example. Thanks for doing the math to prove me wrong!

Presumably you think that the people running the study didn't think of this?

Here's a quote from a New York Times article about the project -

> In the second year of the tournament, Tetlock and collaborators skimmed off the top 2 percent of forecasters across experimental conditions, identifying 60 top performers and randomly assigning them into five teams of 12 each. These “super forecasters” also delivered a far-above-average performance in Year 2. Apparently, forecasting skill cannot only be taught, it can be replicated.

So the answer to the question "What is the probability any one member of the group correctly guesses the result of the next coin toss?" appears to be "reasonably high".

Are you presuming Tetlock et al didn't think of this? That seems pretty unlikely, given what they're studying.

So you're suggesting instead that 'superforecasters' exist, but that they haven't been discovered by finance? It'd be a pretty lucrative market for people able to predict the future. I think it's far more likely that it's less than excellent science.

I'm struck that with a tiny bit of research you could figure out yourself if/how they accounted for the possibility of luck (hint: they did).

Warren Buffet is just a lucky coin flipper.

Warren Buffet does not rely on chance to make his investments.

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