Hair splitting
November 24, 2022 at 1:42 PM by Dr. Drang
A few days ago, Brady Haran at Numberphile released a new video with Ben Sparks. Like most of Ben’s videos, it’s interesting without being super heavy with math. In this case, I think the psychology behind the video is the most interesting part.
The puzzle is this:
What are the chances that there are two people in London with the same number of hairs on their head?
There’s a bit of misdirection in posing the problem in probability terms, as it might lead the listener to think he’s asked about the chances of two randomly selected people in London having the same number of hairs on their heads. A more straightforward—and therefore less tricky—question would be
Are there two people in London with the same number of hairs on their heads?
The answer comes from being able to estimate, within an order of magnitude or so, the number of people in London and the number of hairs on people’s heads. The former is a specific number that’s continually changing, so no one knows it except as a range. And the latter is a range by definition.
I think most people know that the population of London is at least several million. The tougher estimate is of the range of hairs on people’s heads. Ben suggests up to around 100,000, based on a hair density of 100 hairs per square centimeter over a 30 cm × 30 cm area. This hair density is equivalent to a hair spacing of about 1 mm, which seems reasonable to me.
(I’m no hair follicle expert, but any parent who’s gotten a message from their child’s school about instances of head lice showing up in class knows what it’s like to go through their kid’s scalp hair by hair.)
So with the range of hair counts at least an order of magnitude less than the population of London, there have to be at least two people with the same number of hairs. The “chances” asked for is 100%. Puzzle solved.
But Ben didn’t really ask for the solution itself. He wanted your gut reaction—before you did any hair-density calculations. And it’s probably not immediately obvious to most people that the number of hairs on someone’s is well under a million.
This is where the psychology comes in. While watching the video, I thought of a mathematically very similar question, but one that would, I believe, get instant correct answers from almost everyone:
What are the chances that there are two people in London who were born on the same day?
The number of possible birth dates of living people has to be around 40,000, which is within an order of magnitude of the hairs-on-head number. So the answer to this question is also 100%, but I bet most people would answer it correctly without hesitation and without calculation.
Even people who’ve never seen a birth notice in a newspaper probably know they exist. And they know that it’s common for there to be multiple notices every day in a big city. And those who don’t know about birth notices probably know that it’s common for large hospitals to have more than birth per day—and that big cities have many hospitals.
It’s the combination of the familiar—the hair on our heads—with the unfamiliar—how many hairs are there?—that makes Ben’s question interesting. My question, because it’s so easy, wouldn’t be interesting, even though it’s mathematically the same. Good thing Brady has Ben instead of me.
