A probability problem doubt The question is like this:

Determine the probability that in a class of $23$ students two or more students have birthdays on January $1$?

The method to do this is what I am looking for. What is the basic concept which I need to use here?
 A: You start with the (not fully justified) assumptions that


*

*the probability of some student to have birthday on January 1 is $\frac1{365}$ (or should it be $\frac1{365.25}$?)

*the birthdays of the students are not correlated


The first assumption is not justified because nightly room temperature undergoes seasonal oscillations and because doctors may influence the date of birth away from holidays (then again, sometimes everyone wants to have the first baby of the millenium ...), the second is also not fully justified because it might be the case that twins are more likely than not to attend the same course.
But let's ignore these objections. You can compute the probability that none of the students is born of january 1, and you can compute the probability that exactly one is born on January 1. From this you can obtain the fnal result.
A: The question can be answered by using binomial distribution. Let $X$ be the random variable that denote the number of students to have birthday on January $1$. Let $p$ be the probability of some student to have birthday on January $1$ and $p=\dfrac{1}{365}$ as stated by Hagen von Eitzen (Assuming $1$ year = $365$ days). The easiest way to obtain the probability of two or more students have birthdays on January $1$, $\Pr[X\ge2]$, is obtaining its complement first, the probability of less than two students have birthdays on January $1$, $\Pr[X<2]$.
$$
\begin{align}
\Pr[X<2]&=\Pr[X=0]+\Pr[X=1]\\
&=\binom{23}{0}p^0(1-p)^{23}+\binom{23}{1}p^1(1-p)^{22}\\
&=\binom{23}{0}\left(\frac{1}{365}\right)^0\left(1-\frac{1}{365}\right)^{23}+\binom{23}{1}\left(\frac{1}{365}\right)^1\left(1-\frac{1}{365}\right)^{22}\\
&=\binom{23}{0}\left(\frac{364}{365}\right)^{23}+\binom{23}{1}\left(\frac{1}{365}\right)\left(\frac{364}{365}\right)^{22}\\
&\approx0.99817.
\end{align}
$$
Thus, the probability of two or more students have birthdays on January $1$ in a class of $23$ students  is
$$
\Pr[X\ge2]=1-\Pr[X<2]=1-0.99817=0.00183.
$$
