Blink Blink

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%.

——

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.

2 thoughts on “Blink Blink

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