Here’s How To Take A Blink-free Group Photo

Whenever a group photo is taken, it always seems that there’s at least one person that blinks. So the question that naturally arises would be: how many photos should be taken so that we get at least o

29 Scientists at the Solvay Conference in 1927

The other day I was reading this article about the Solvay Conference, probably the most intelligent picture ever taken, 1927. 17 of these 29 scientists in the photo later were or later went on to become Nobel prize winners, including Marie Curie, the first female Polish scientist who was awarded the Nobel prize in two scientific disciplines.

But that’s not the point of this article. Instead I want to figure out how many photos did they take to get at least one when no one was blinking.

Try looking through some old group photos with lots of people, perhaps an old high school reunion photo or a birthday part photo, did you find someone that blinked?

Don’t be surprised if you did!

Because as the mathematics here will show that it’s more common than you think. And at the end, we can find an estimate of the number of photos taken so that none of the scientists at the Solvay Conference was blinking.

---

Let’s Lay Out Our Model

According to the internet, the average blink rate of adult is around 15 to 20 times per minute. And when we focus on screens, our blink rate tends to drop to around 5 to 7 per minute.

But for the sake of modelling this as a mathematical problem, let’s assume that the average person blinks about 15 times per minute (≈ one blink every 4 s). We denote r as the blink rate.

Modern cameras (or phones in burst mode) have an effective ‘shutter‐open’ window. We will label that length T.

In practice, T is usually on the order of 0.05–0.10 s (50–100 ms).

Now blinking can be thought of as a series of random, independent events occurring at a roughly constant average rate. In statistics, events that meet the following three criteria are well‑modeled by a Poisson process.

• Memoryless: the chance to blink in the next moment doesn’t depend on how long it’s been since the last blink.

• Constant rate: over short intervals, your blink frequency r remains approximately constant.

• Independence: each blink is independent of everyone else’s.

Under a Poisson model, the probability you don’t blink during a window of length T is

For example if the blink rate r is 0.25 and shutter-open window is 0.1, the probability that you don’t blink during that time frame of 0.1 seconds is very high.

---

An n-person Group

The beauty of mathematics lies in its ability to go from the specific to the general. In our case, we will extend our probability of no blinks from one person to an arbitrary group of people.

We will assume each person blinks independently, the chance all n people keep their eyes open in that shot is

So in a group of 10 people where the blink rate r is 0.25 and the shutter-open window is 0.1, the probability that no one blinks will drop quite a bit.

So far so good.

But what we are really after is how many shots we should take to make sure no one is blinking.

Or perhaps precisely, how many shots must we take to achieve a desired confidence level.

Let’s see.