## Unamazing Statistic

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.)

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### 4 Responses to Unamazing Statistic

1. Eric says:

Agree; I would be curious to know what assumptions led to the stated probability, referred to in the article as “the odds of three generations of the same family all having boys born on the same day.” Interestingly, 272911 is prime! (As you know, “odds” is often used misleadingly; to say that the odds are 272910:1 is to say that the probability is 1/272911.)

Conditioned on observing a particular existing grandfather/father/son triple, I agree that the probability should be around twice the stated value. (This despite the non-uniform distribution of birthdates throughout the year; this actually only makes these sort of “coincidences” more likely, not less.) But the reference to “having boys” suggests that the intended probability is that of a man having a son and grandson all with the same birthdate, which requires assumptions about distribution of family size, clarification of whether the younger two must be father and son or may be uncle and nephew, etc.

But even assuming, for example, that every family has two children, with boys and girls equally likely, and birthdates uniformly distributed, the probability is still very near 1/365^2. The only other reasonable assumption I can think of that significantly changes this answer is to have *one* child per family… but then the probabiilty is off by a factor of two in the *other* direction. I don’t get it.

2. nate 519 says:

i just had a child who was on my bd and i was also born on my father’s bd threee generations all the same day

3. nate 519 says:

oddly enough my brother has a daughter born on his bd and my sister was born on her mothers bd