I was posed a question:

What are the odds that in the span of the one eighth of a second it takes for a camera shutter to open and close, 18 people will all have their eyes open at the same time?

ANSWER: Assuming a blink rate of 20 blinks per min with an average blink duration of $\frac{1}{4}$ second, the probability is 12.2%, or roughly 7 to 1 against. If the people make a reasonable effort to keep their eyes open, they may affect their blink rate. As an example, if the average blink rate can be subdued to a mere 4.75 blinks per minute, the probability rises to 59%, or roughly 1.4 to 1 in favor.

METHOD: In order to solve this problem, I accepted that the human eye blinking is a Poisson Process thus distributed according to the Poisson Distribution. In other words, if we let the average number of blinks over some interval of time be given by $\lambda$, then the probability of $x$ blinks occuring over that interval is given by

$\displaystyle P(x) = \frac{\lambda^x e^{-\lambda}}{x!}$

Now, because the shutter is open for only $\frac{1}{8}$ of a second and a blink will last $\frac{1}{4}$ of a second, then if an individual blinks anywhere from $\frac{1}{4}$ of a second before the shutter opens to when the shutter closes, they will appear to have their eyes closed in the picture. So we are look at an interval of $\frac{1}{4}+\frac{1}{8} = \frac{3}{8}$. Thus,

$\displaystyle \lambda =\left( 20 \mbox{ blinks per second } \right) \left( 60 \mbox{ sec. per min. } \right) \left(\frac{3}{8} \mbox{ sec.} \right) = \frac{1}{8}$

Therefore the probability that a single person’s eyes will be closed is$\displaystyle P(1) = \frac{1}{8} e^{-1/8} \approx 0.1103$. Thus the probability that their eyes are open will be $1-P(1) \approx 0.8897$. Assuming independence among our 18 (no one person blinking affects another person’s blinking), the probability of all 18 having their eyes open will be $(0.8897)^{18} \approx 0.122 = 12.2%$.

If we repeat this process changing only the 20 blinks per minute to 4.75 blinks per minute, we obtain 59%.

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Now wasn’t that special. HT: jonboy, for the question.

Any more questions, fire away in the comments or send me an email. I love to be distracted from actual work I need to be doing.