# More than 100% pregnant

May 9, 2024 at 11:21 PM by Dr. Drang

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?

Remarkably, the answer is not

$$n\left(\frac{1}{n}\right)=1$$It’s

$$1-{(1-\frac{1}{n})}^{n}$$The term in the parentheses is the probability of it *not* happening in a single trial. That term raised to the *n*th 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

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

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