# Odds of births of four generations [closed]

What are the odds of the following:

Great Grandmother was born January 14 (1/14)

My Grandfather was born April 11 (4/11)

I was born on January 14 (1/14)

My son was born on April 11 (4/11)

FYI: My Great GrandMother lived to be 100 years old. She passed just a few years ago.

• Assuming it has happened, the answer is 100%. It is incredibly unlikely that anyone would formulate such a set of conditions before making a particular observation, which is the only context in which a question about probability has a meaningful answer. Thus, a more relevant question would be "what are the chances that somebody who looks for numerical patterns in birth days and months would come up with some relationship among four of their kin that they find interesting?" That, too, is unanswerable, because we cannot determine what patterns would attract your attention.
– whuber
Commented Jul 7, 2014 at 19:55
• It might be a good idea to look at David J. Hand's book "The Improbability Principle:Why Coincidences, Miracles, and Rare Events Happen Every Day". Published by Scientific American 2014. Commented Apr 24, 2017 at 0:07

As whuber points out, it doesn't mean a whole lot to ask about the probability of something after it's already happened. But in the spirit of your question, we could ask "What's the probability that if we pick 4 random people then 2 of them will share a birthday and the other 2 will also share a birthday?"

Here, it depends on whether you assume these events are independent and if it's equally likely to be born on any day of the year. If we assume that any day is equally likely (I don't think this is strictly true, but it's a reasonable assumption to make), and that these 4 birthdays are all independent of each other (and we ignore leap years), then the chance of this happening is just:$$({\frac 1{365})^2}$$

This is true because person 1 can be born on any day, then there's a $\frac 1{365}$ chance that person 2 shares that birthday. Same goes for persons 3 and 4 (the calculation would be a little different if you specified that you wanted the two days to be different days). This may not quite fit with what you're really trying to ask - if you specify exact dates beforehand and then pick 4 random people the chance would be $$({\frac 1{365})^4}$$

However, this is misleading as this is the probability of ANY combination of $4$ birthdates. In other words, it's also the chance that if you take $4$ people their birthdays will be 1/1, 1/1, 1/1, and 1/1. Or 1/1, 1/2, 1/3, and 1/4. OR 1/2, 3/4, 5/6 and 7/8. Or something with no discernible pattern like 1/7, 10/3, 4/23 and 7/31.

• Actually, this is not correct. You give the chance for four people all having specific birthdays (e.g., twice Jan 14 and twice Apr 11). This is not the same as "two out of four persons share a birthday, and the other two share one, too", where we do not specify in advance which days these are. Person A is free in "choosing" his birthday. Person B has the same birthday as A (chance of 1/365). C is again free, and D is determined by C (again chance of 1/365). Overall answer to your problem: (1/365)^2, not ^4. (Of course, the OP may have yet another problem in mind.) Commented Jul 7, 2014 at 20:09
• Oh, good catch. Will update. Commented Jul 7, 2014 at 20:11
• This noble attempt at an analysis (+1) shows just how vague the question really is. By positing an interpretation that, although consistent with the problem statement, is just one of many other possible consistent interpretations, it highlights one of the many epistemological difficulties lurking here.
– whuber
Commented Jul 7, 2014 at 21:36