I think what he meant was that if Tao indeed has an IQ of 230, then on average one seven-billion-people-world out of every 57 million such (random) worlds contains one person as smart as Tao, or, that you need to sample something like 400 quadrillion people to find one such person. So your "122" figure doesn't seem to have any meaning here (at least to me).

Yeah, that's what I meant. (Except that the world population is now nearly 8 billion.)

Sometimes the population of the world seems like the price of Bitcoin to me. Wasn't is 7 billion just yesterday?

Yes, don't worry it will probably level at 11 billion

No, that was this morning. Yesterday it was six, and the other day four.

I think the parent's point is that you would need 50 something millions worlds like ours to find a single person with such a high IQ score.

I haven't checked the maths, but I know the Gaussian distribution is falling fast past a few standard deviation, so such a result would not surprise me.

I admit I'm not very informed about what IQ denotes, but wouldn't such a probability need to include the population of the world? Is that a parameter in the calculation of IQ?

Allow me to inform you.

IQ stands for "intelligence quotient".

It was developed as a ratio between mental age and physical age, times 100. So if you performed as expected, your IQ was 100. It was originally developed as a way of finding people who were behind, literally "retarded". However it also proved useful as a way of finding people who are intelligent as well.

After IQ tests were adopted in school systems, we found that they were approximately normally distributed with a mean of 100 (by definition), and a standard deviation of 15-16. For a variety of reasons (including identifying military recruits to train for specialized roles), there was a desire to have ability tests aimed at young adults. We developed those and scaled them to be explicitly normal with a mean of 100 and standard deviation of 15 or 16 (depending on the test). We also found that childhood IQ is a fairly good predictor of your later adult abilities, and therefore many of those tests are ALSO called IQ, even though there is no quotient involved.

On the adult tests, the tests do not scale out to IQ 230, nor is it likely that anyone is that many standard deviations out. But on the child tests, there is no problem scaling it. And it turns out that, in practice, the tails are heavier for the original type of IQ test. Which means that it is more common to find a 6 year old who performs at a 14 year old's ability, than a person who is over 8 standard deviations out.

Terry Tao had his IQ officially measured on a childhood test, not an adult one. However there has been no test since that is able to reliability measure his ability, particularly in math.

Consider, at 7, his SAT score put him well within the top 1% of college-bound kids.

At 10, his performance on the International Math Olympiad meant that he was literally in the top handful of high school students in the world.

Yeah, we don't have properly scaled tests for that.

I'd add on top of that that these days adult IQ is instead defined to have a normal distribution with mean 100 and standard deviation 15. This is achieved by 'grading it on a curve'.

That's why my above comment was explaining why the 230 figure couldn't possibly make sense under the modern definition.

An adult test can't give such a score.

However the Stanford-Binet test is still used for children. And in young children can still produce extremely high IQ scores.

Also note that while usually the standard deviation is made to be 15 these days, there are still tests, like the Binet and OLSAT, where the standard deviation is 16. That's why I said that the standard deviation depends on the test.

Theoretically not; IQ is ostensibly a statistical distribution, so it applies regardless of population size.

Yes, I used the world population figure when I was calculating that probability. But the world population size is not used in the definition of IQ itself.