The drudgereport carried an article yesterday about a grandfather, father, and son all with the same birthday; with the headline “what are the chances of that?” The article claims the odds to be 272,910 to one. My math comes up a little different.
The chance of the grandfather and father having the same birthday are 365-to-1 (ignoring leap years). Given this, the probability of father and son with same birthday is also 365-to-1. Assuming these are independent events, that puts the odds somewhere around 133,225-to-1. So, on the same order of magnitude, but about twice the stated probability.
Even so, this is just the odds of a particular family observing this occurrence. The odds of any family observing this are (1-(1-p)^N) where p is 1/133225 and N is the number of grandfather-father-son groupings.
If there are 100,000 such groupings (probably a low estimate, but I’m not that familiar with British population dynamics – these guys were British, by the way), then the probability of at least one grandfather-father-son group all having the same birthday is just under 53%. If there are 1,000,000 grandfather-father-son groups, then the probability of at least one having all the same birthdays is around 99.95%.
So, this doesn’t appear to be an outlandish occurrence at all.
(I know, I usually write about politics and religion – but the math question here seemed interesting.)