[Q] what are the odds

[uncle-duke]

So if a man has a son on his birthday the chances are 1in365, foregoing leap years. If that son grows and has a son on his birthday it stays a 1in365. Now for the start of this chain it became 1 in 365 squared. I have been attempting to figure adding the spacing between generations of a fixed setting.

So our first gen is twentynine on the birthday his son is born. And the second gen continues the trend on his twentyninth birthday. What becomes the third generations chances to replicate this trend?

 

Now I ask because this is true in my life. My father and I as well my son share a birthday twentynine years seperated. I have mulled this a number of times but the years varibility throws me when we consider the adult males span of procreation.

So the discision of off spring becomes a variable but we might assume that evolution has a helping hand and the progression continues. As for male or female it is neither nor to me, but I think it complicates the equation with a multiple of two.

 

So what am I just trying to manage statistics with too little knowledge?

[Moontanman]

If you assume the birthday is completely random then each time the odds are the same. Birthdays can be manipulated both by induced child birth and by planning the pregnancy and the birthday to within a a couple of weeks so it's no longer 365 to one. It does seem very unlikely grandpa, dad and grandson would all share the same birthday. Not sure about real odds though.

[CraigD]

Assuming each child’s day of birth and gender are fairly random – that is, that parents attempting to force births to occur on a specific date thought such means as timing conception, inducing labor, or c-sections – and assuming the parents have only sons, or continue having children until they have one and only one son, your calculations are nearly right, uncle-duke.

 

The probability of a father and only son having the same birthday given the above is about [math]\frac{1}{365}[/math]. The odds of this repeating for the next generation is [math]\frac{1}{365^2}[/math], for the next generation, [math]\frac{1}{365^3}[/math], and so on. So, given the current world male population, and assuming everyone follows the above assumption, there should be about 8 million sons with their father’s birthday, 22 thousand with both their father’s and grandfather’s, only 62 with father’s, grandfather’s and great-grandfather’s, and none with father’s, grandfather’s and great-great-grandfather’s.

 

Changing the assumptions can make the calculation more complicated. For example, if each generation keeps trying for a birthday match their chance of success is about [math]1-\left(\frac{364}{365}\right)^n[/math], where [math]n[/math] is the father’s number of male children. If, say, the line of fathers are polygamous monarchs with, say, 100 sons, the probability per generation is about 24%.