Birthday Paradox/General/3

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Theorem

Let $n$ be a set of people.

Let the probability that at least $3$ of them have the same birthday be greater than $50 \%$.


Then $n \ge 88$.


Proof

Let $\map F {r, n}$ be the number of ways to distribute $r$ objects into $n$ cells such that there are no more than $2$ objects in each cell.

Let there be $d$ cells which are each occupied by $2$ objects.

These can be chosen in $\dbinom n d$ ways.

There remain $s = r - 2 d$ objects which can then be distributed among $n - d$ cells in $\dbinom {n - d} s$ ways.

In each such arrangement, the $r$ objects may be permuted in:

$\dbinom r 2 \dbinom {r - 2} 2 \cdots \dbinom {r - 2 d + 2} 2 \paren {r - 2 d}! = \dfrac {r!} {2^d}$

different ways.

Hence:

$\map F {r, n} = \dbinom n d \dbinom {n - d} s \dfrac {r!} {2^d}$


So the probability of exactly $d$ pairs and $s$ singletons, where $d - s \le n$, is given by:

$\dfrac {\map F {r, n} } {n^r}$

If we assume a $365$-day year, we have that the probability that at least $3$ of them have the same birthday is given by:

$\map \Pr r = 1 - \displaystyle \sum_{d \mathop = 0}^{\floor {r / 2} } \dfrac {n! \, r!} {n^r 2^d d! \paren {r - 2 d}! \paren {n + d - r}!}$

where $n = 365$.

We require the smallest $r$ for which $\map \Pr r > \dfrac 1 2$.

The result yields to calculation.

$\blacksquare$


Sources