Example 16. At the faculty, 500 students are enrolled. What is the probability that September 1st is the birthday of $k$ students of this faculty? Calculate this probability for the values $k=0,1,2,3$.
Solution. Since $n=500$ is quite large and $p=1 / 365$ is quite small, we can assume that the random number $X$ of students born on September 1 follows a Poisson distribution with parameter $a=n p=500 / 365=1.36986$.

Using formula (3.3.1), we get $p_{0}=P(X=0)=e^{-a} \approx 0.2541$. Using formula (3.3.8), we sequentially find:

$$
\begin{aligned}
& p_{1}=P(X=1)=\frac{a}{1!} e^{-a}=a p_{0} \approx 0.3481 \\
& p_{2}=P(X=2)=\frac{a^{2}}{2!} e^{-a}=\frac{a}{2} p_{1} \approx 0.2385
\end{aligned}
$$

$$
p_{3}=P(X=3)=\frac{a}{3} p_{2} \approx 0.1089
$$