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Someone brought up in conversation that three of her friends had consecutive birthdays (such as November 10, 11, and 12), and I wanted to figure out how likely that is for any randomly selected three people, assuming that birthdays are randomly distributed and the birthdays of two people in a sample are independent. My answer:

= possible arrangement of consecutive birthdays / possible arrangements all birthdays
= 365 / 365^3
= 0.0000075 

Does that sound about right? Or am I missing something?

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    $\begingroup$ Your problem is ill-posed. The answer depends on the number of people the 3 were randomly selected from. $\endgroup$ – Michael R. Chernick Sep 10 '12 at 3:18
  • $\begingroup$ combinatorics/probability $\endgroup$ – pyCthon Sep 28 '12 at 17:21
  • $\begingroup$ My daughter, her father and brother are right in a row April 16,17,18..im curious if there is anybody else like this? Brother is 16th, dad is 18th and daughter right in the middle 17th $\endgroup$ – Rachelle Sep 3 '17 at 20:55
  • $\begingroup$ @Rachelle first, this is not an answer so I turned it into a comment. Second, it is unrelated to statistics. Your question is basically: is it possible that a strange coincidence will ever happen? Yes, they happen all the time. $\endgroup$ – Tim Sep 3 '17 at 22:07
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For simplicity, ignore leap days and that the distribution of birthdays is not uniform.

There are $365$ sets of consecutive triples of days. We can index them by their first day.

There are $3! = 6$ ways the $3$ people can have a particular triple of distinct birthdays.

There are $365^3$ ways the people can have birthdays, which we are assuming are equally likely.

So, the chance that three random people have consecutive birthdays is $\frac {6 \times 365}{365^3} = \frac {6}{365^2} \approx 0.0045\% \approx 1/22,000.$

Of course, if you have $60$ friends, there are ${60 \choose 3} = 34,220$ ways to choose $3$ of them, and so the average number of triples with consecutive birthdays among your friends is about $1.5$, even if you disregard the chance that the real pattern was a superset such as "consecutive or equal" or "within 2 days of each other." If this is counterintuitive, look up the Birthday Problem.

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  • $\begingroup$ I had doubts about the necessity of the 3! term, so I wrote a program to select three random number between 0-364 and test if they are consecutive (including wraps). Roughly 1:22000 accurately represents my results. $\endgroup$ – Octopus Aug 21 '17 at 16:47
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    $\begingroup$ In hindsight, the 3! explains why I had to sort the order every time I tested. $\endgroup$ – Octopus Aug 21 '17 at 16:57

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