covering the birthday bases

Question

The standard birthday question is, how many people have to be in a math class for the probability of two having a shared birthday is > 1/2. With standard assumptions about birthday distributions and independence of birthdays.

A slightly different question.

How many people need to be in a stadium for the probability to be at least 1/2 that for each day of the year, two people in the audience share that day as their birthday. Two part question: (a) exclude 29 February. (b) include 29 February.

Can you estimate N in terms of p? I.e. question as posed has p = 1/2. What if we want p >= 3/4?

Answer 1

An estimate for a) would be

$p \approx ( \sum_{k=2 .. n} \mathcal{B}_{n, 1/365}(k) )^{365}$

$\mathcal{B}_{n,p}$ is the binomial distribution

The solution would have to be found numerically

What makes this equation an approximation is that the distributions $p(n_{people})$ which are marginally identical for each day of the year are not independent. E.g., given that all people in the stadium are born in 1st of January you know for sure that none were born in the other days, so the product is false

You can also write out your binomial terms to get this

$p \approx (1 - (\frac{364}{365})^n - \frac{n}{365}(\frac{364}{365})^{n-1})^{365}$

Answer 2

The chance that there are at least two people born on Jan 1 is $1-\left(\frac {364}{365}\right)^N-N\frac {364^{N-1}}{365^N}$. If we pretend that the days are independent, which is not quite fair but will not be too far off, we want $$\left(1-\left(\frac {364}{365}\right)^N-N\frac {364^{N-1}}{365^N}\right)^{365}=\frac 12\\1-\left(\frac {364}{365}\right)^N-N\frac {364^{N-1}}{365^N}\approx 0.9981\\\frac {(364+N)364^{N-1}}{365^N}\approx 0.0019$$
Alpha gives $N\approx 3106$

For larger $p$, just plug that in. The $0.9981$ is an approximation to $\left(\frac 12\right)^{1/365}$

The chance that two people were born on Feb 29 is $1-\left(\frac {1460}{1461}\right)^N-N\frac {1460^{N-1}}{1461^N}$, which is about $0.3740$ when $N=3100$ so you need some more. It would take exploration with Alpha.