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 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 , then the probability of blinks occuring over that interval is given by
Now, because the shutter is open for only of a second and a blink will last of a second, then if an individual blinks anywhere from 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 . Thus,
Therefore the probability that a single person’s eyes will be closed is. Thus the probability that their eyes are open will be . 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 .
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.