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I understand the solution to the birthday paradox. But I was wondering how I would calculate the probability of two people having the same age, or the exact birthday, down to matching years.

I am thoroughly confused. Please help.

warpcore
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  • You'd have to start with some assumption on the ages of the people in the group. Can they be up to 100 years old? 1000 years old? – Austin Mohr May 07 '13 at 18:33
  • You need to define your question more precisely, especially because you say "down to matching years." What interval of ages/years are we willing to consider? – xisk May 07 '13 at 18:35
  • "birthday, down to matching years" $: \mapsto :$ birthdate $;$ ? $;;;$ –  May 07 '13 at 18:59
  • How much approximation do you want? Here are some relevant statistics:

    Step 1 (Distribution of Birthdays in a Year): http://www.panix.com/~murphy/bday.html

    Step 2 (Age Distribution by Country): http://www.indexmundi.com/facts/visualizations/age-distribution/

    Step 3 ("Lifespan Depends on Month of Birth"): http://www.pnas.org/content/98/5/2934.full

    And so on...

     – CommonerG May 07 '13 at 19:52
  • On second thought, just assume everything is uniform (or the room is in the neonatal ward of a hospital). – CommonerG May 07 '13 at 19:59

1 Answers1

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The simplest form of this is to assume that birthdays are uniformally distributed over the $365$ (non-leap) calendar days and that everyone in the room was born in a $Y$ year time-span with birth year again being uniform (and independent of birth day). It follows that birthdates are uniformally distributed over the $365*Y$ possible dates. The end result looks a lot like the standard birthday problem analysis:

If the room has $N$ people, the probability no one shares the same birthdate is:

$$\frac{1}{(365*Y)^N}\frac{((365*Y)!}{((365*Y)-N)!}$$

And so the probability that at least two share the same birthdate is:

$$1-\frac{1}{(365*Y)^N}\frac{((365*Y)!}{((365*Y)-N)!}$$

This approximation will fail to represent real probabilities so far as birth days are not uniform, birth years are not uniform, and in fact birth day and birth year are not independent (see comments).

CommonerG
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