problem

The original birthday problem, also known as the birthday paradox, asks how many people need to be in a room to have a 50% chance that at least two have the same birthday. Since this number is lower than what our intuition tells us, it is sometimes also considered a paradox.

Generalizing this problem, it is about a set of $D$ objects (e.g. days of the year) from which we draw $N$ samples uniformly at random with replacement (e.g. birthdays) to determine the probability $\bar{P}$ that all samples are unique. This can be calculated with

$$\bar{P}(D, N) = \frac{(D)_N}{D^N}$$

where $(D)_N$ $=$ $D$ $*$ $(D-1)$ $*$ $(D-2)$ $*$ $...$ $*$ $(D - N + 1)$ is the number of ways $N$ *different* items can be chosen from $D$ and $D^N$ is the number of ways *any* $N$ items can be chosen among $D$ (assuming the silent, potentially incorrect, assumption that all items are uniformly distributed in $D$). The probability $P$ of the existence of non-unique samples is then the complement $P = 1 - \bar{P}$.

Can you guess the answer to the original birthday problem? Go ahead and put your guess to test below!

Use the calculator below to calculate either $P$ (from $D$ and $N$) or $N$ (given $D$ and $P$). The answers are calculated by means of four methods. When calculating $P$, three different methods are used by default whereas only one is available for calculating $N$. The trivial method is used whenever trivial problem instances are encountered. When calculating $N$, the results are ceiled since no fraction of a sample may be taken. The result therefore reflects the needed size of $N$ to have a probability, equal to or higher, than the input $P$.

Some limitations exist and a single method has at most 1 second to solve a problem instance. The size of the input is also limited. In case of a high load, the server may choose to deny requests temporarily.

accepted

accepted

Resulting input data:

$\begin{align*}
\rule[-8px]{0pt}{30px} D &= 365\\[-4px]
\rule[-8px]{0pt}{30px} N &= 23
\end{align*}$

accepted

accepted

Resulting input data:

$\begin{align*}
\rule[-8px]{0pt}{30px} D &= 365\\[-4px]
\rule[-8px]{0pt}{30px} P &= 0.5
\end{align*}$