September 15, 2008 | 43 CommentsHere is a question I often give on exams:

What is the probability that the next child to be born will be a genius? Give me a number and fully explain your answer.

There is not, of course, a single correct answer. What I just said is an important point, so let’s not skip lightly over it: there is no correct answer; at least, there is no way anybody can know the correct answer.

That nobody can know with certainty answers to questions of this type is under appreciated. I want people to learn this because we are, as I often say, too sure of ourselves.

What I want to see in the answer is acknowledgment of the ambiguities. First, what is a genius? Surely that word is overused to a remarkable extent. For example, this list says, with a straight face, authoress JK Rowling and movie maker Stephen Spielberg are geniuses. I often have the idea that to *not* call some eminence a genius is nowadays taken as a slight. However, a moments’ thought suffices to show that people exaggerate—if you are willing to take that moment.

The next step is to think of some geniuses for the sake of comparison. It’s best to think of dead ones so that you are not overly influenced by current events. After all, only history can truly judge genius. If you agree with even part of this, you will have made the next most important step: admitting that you can be biased.

How about some dead geniuses? Einstein pops into nearly everybody’s head first. Then, for me, Mozart, Beethoven, Shakespeare, Newton, and the guy who invented beer. No, I’m not joking about that last name. The point is my historical knowledge is modest, and most of the names I pick are men from the last 500 years, and most are from Western culture. Humanity is older than 500, of course, and there are other cultures besides our own, so I know that my knowledge of who is a genius is limited. That’s what got me to thinking about the brilliant soul who invented beer. He did so, probably in Sumer, before people wrote down incredible deeds of this sort.

This line of thought eventually leads to other cultures (Confucius, maybe Lao Tzu) and other times where writing was non-existent (was there just one person responsible for the wheel and agriculture?). There must be a lot of geniuses I don’t know, and some that nobody can ever know.

Next step is to count, and to acknowledge that exact counting is an impossibility. Still, we can count to the nearest *order of magnitude*. This means “power of 10”, and it represents an enormously popular method of approximation. If you can get your answer to within “an order of magnitude” (a power of 10), you are doing good. The first power of 10, or 10^{1}, is just 10. The second power is 10^{2}=100, and so on.

So how many geniuses? Certainly more than 10, definitely less than 10,000, or the 5th order of magnitude. Could there have been a 100 geniuses? Given my above list, I say yes. 1000? I’m less likely to believe this number, but since I have said that there were lots of geniuses who went unsung, I can’t exclude it. Still, an order of magnitude more than this seems too large.

We have done a lot so far, but we still haven’t answered the question “What is the probability that the next child to be born will be a genius?” The answer will look something like # of geniuses who have ever lived / # of people who ever lived. Coming to *this* equation is crucial. This is because the question implies—I emphasize, it does not explicitly state—we are asking a question about all humanity. And all humanity certainly means all humans who have ever lived.

Thus far, we have nailed down the numerator in this equation to the nearest order of magnitude or so (10^{2} to 10^{3}). How about the denominator?

What evidence do we have? Well, there about about, to an order of magnitude, 10^{10} or 10 billion people alive today. 100 years from now, nearly of these people will be dead and a new set, probably the same order of magnitude will take its place. Anyway, 100 billion people alive 100 years from now feels way too large to me, and 1 billion way too small, especially given recent population trends.

100 years ago, there were about an order of magnitude less people alive (nearly all of them different from the set we have today), or about 10^{9} or 1 billion. How many 100s of years can we go back? About 2000, since the best guess is humanity arose about 200,000 years ago. That’s close enough; it’s within an order of magnitude. Without doing any math—just going by gut—we can guess that adding today’s 10^{10} to last century’s 10^{9} (11 billion so far), and to the previous 199 centurys’ diminishing contributions (each previous century had fewer people), we arrive at about 10^{11}, or 100 billion.

Was that larger than you had first guessed? This number usually surprises most people. But having a guess gives us our denominator as that we can finally solve our equation, which is

10^{2}

—— = 10^{-9}

10^{11}

of, if there were 10^{3} geniuses, 10^{-8}. In words, it’s anywhere from 1 in a billion to 1 in 100 million.

Not very good odds, right?

This was a lot of thinking for such a simple question, wasn’t it? If you would have written down, as student’s often do, an answer “1 in a 100” or “1 in 1000” you would have got the answer wrong. Both answers imply that we should be flooded with geniuses, an answer which no observation supports.

Of oft-heard complaint among professors is that students don’t think about the answers they give. I agree with this, but I think it’s more than just students. It holds for professors and ordinary civilians, too.

“1 in a 100” is absurd, and far too certain. Just a few moment’s thought shows this. How many answers that we give in life are just as absurd?

Some kids will write, “I don’t know.” I usually give them 1 point for this because, after all, it is the strictly correct answer. But that answer is too certain itself. We *do* know something about the answer and we can answer it partially. We should always quantify uncertainty in any question and not seek the easy way out by given answers that are too certain.

Here, for fun, is another question I give:

How many umbrellas are there in New York City?