What is the probability of 2 or more people having the same birthday in a class of 25?

The probability that none have the same birthday is (364/365)*(363/365)*(362/365)*...*(341/365) ≈ 0.431

 

So the probability that at least 2 have the same birthday is 1 - 0.431 = 0.569

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It is 1- the probability that they all have different birthdays

 

$$\\=1 - (\frac{365}{365}*\frac{364}{365}*\frac{363}{365}*\frac{362}{365}*\frac{361}{365}*......*\frac{366-25}{365})\\\\
=1- (\frac{365}{365}*\frac{364}{365}*\frac{363}{365}*\frac{362}{365}*\frac{361}{365}*......*\frac{365-24}{365})\\\\
=1- (\frac{365P24}{365^{25}})$$

 

$${\mathtt{1}}{\mathtt{\,-\,}}{\frac{{\left({\frac{{\mathtt{365}}{!}}{({\mathtt{365}}{\mathtt{\,-\,}}{\mathtt{25}}){!}}}\right)}}{{{\mathtt{365}}}^{{\mathtt{25}}}}} = {\mathtt{0.568\: \!699\: \!703\: \!969\: \!463\: \!9}}$$

 

So P(2 or more have the same birthday) is  approx 0.57      or    57%