# More than 100% pregnant

Kieran Healy tweeted out a gem earlier this evening:

Yes, Professor Huberman, a probability of 120% is indeed “a different thing altogether.”1

While I find it fascinating that someone can run through an argument that clearly proves they are wrong and continue along as if they hadn’t, I don’t have any expertise in psychology, so I won’t try to explain it.

I do, however, have some expertise in probability, and Huberman is dancing around the edge of one of my favorite problems: If there’s a 1 in n chance of something happening in one try, what’s the probability of it happening in n tries?

$n\left(\frac{1}{n}\right)=1$

It’s

$1-{\left(1-\frac{1}{n}\right)}^{n}$

The term in the parentheses is the probability of it not happening in a single trial. That term raised to the nth power is the probability of it not happening in n trials (assuming independence between trials). And subtracting that from 1 gives us the probability of it happening (at least once) in n trials.

What’s great about this problem is that you can make a pretty decent estimate of the answer without doing any calculations. The reason is

$\underset{n\to \infty }{\mathrm{lim}}{\left(1-\frac{1}{n}\right)}^{n}=\frac{1}{e}\approx 0.37$

and so

$\underset{n\to \infty }{\mathrm{lim}}1-{\left(1-\frac{1}{n}\right)}^{n}=1-\frac{1}{e}\approx 0.63$

Now, in the Huberman problem n is 5, which is not especially close to infinity, but luckily the convergence to the limit is pretty rapid.

I would have guessed the answer for $n=5$ was about 0.65; it’s actually 0.67. Not super accurate, but better than 1.

Update 10 May 2024 12:21 PM
John D. Cook also saw the Huberman clip and wrote a nice post this morning about how adding probabilities is a reasonably close approximation if both

• the base probability for a single trial is small; and
• the number of trials is small.

Absolutely true, but not as much fun as a solution that ends up with Euler’s constant popping up out of nowhere.

1. The clip is from this 4 hour, 22 minute (!) video episode of the Huberman Lab Podcast. Imagine if Huberman were John Gruber’s guest on The Talk Show