In the original “Birthday Problem”, the year is divided into 365 discrete days, and a “collision” is defined as being born on the same calendar day. Using this definition, two people can be born two minutes apart, but this will not be called a collision because one was born before midnight and the other after midnight. My question is: If we define a collision as being born less than 24 hours apart (possibly in different years), what is the probability of no collisions in a group of m people? I am assuming that the time of birth T (measured in days) is a continuous random variable with a uniform distribution between 0 and 366 (366 is actually 0 for the next year). Clearly an upper limit on this probability is the classic solution for the birthday problem. Can a lower limit be found? Maybe an approximation? Of course I can try simulation, but an analytic solution is better since it can be applied to several engineering problems such as the occurrence of defects on a camera sensor, which cause the picture to be blurred if they happen to be too close together (this is actually the problem I am interested in solving – if there are m randomly distributed defects, what is the probability that no two are close together).
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Your question is equivalent to $\Pr(|X-Y|<1)$, where $X$ and and $Y$ are independent and following the uniform distribution $U(0,365)$. – user158565 Oct 11 '18 at 21:46
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This would be correct for 2 people, but what is the answer for m people? – Zahava Kor Oct 12 '18 at 23:42