<s>[INST] what is the probability that drawn balls have the same number? In the urn, there is one ball with number $1$, $2$ with number $2$, and so on until $n$ balls with the number $n$.  
From the urn, we draw two balls. 
Calculate the probability that the two drawn balls have the same number.
 [/INST] Overall there are $N=1+2+\cdots+n=\frac{n(n+1)}{2}$ balls.
You need to calculate:
$$\begin{align}P(2\cap 2)+P(3\cap 3)+\cdots +P(n\cap n)=\\
\frac{2}{N}\cdot \frac1{N-1}+\frac{3}{N}\cdot \frac2{N-1}+\cdots+\frac{n}{N}\cdot \frac{n-1}{N-1}=\\
\frac{1}{N(N-1)}\sum_{k=2}^n k(k-1)=\\
\frac{1}{N(N-1)}\sum_{k=1}^n k(k-1)=\\
\frac1{N(N-1)} \left[\sum_{k=1}^n k^2-\sum_{k=1}^nk\right]=\\
\frac1{N(N-1)} \left[N\cdot \frac{2n+1}{3}-N\right]=\\
\frac{2(n-1)}{3(N-1)}=\\
\frac{4(n-1)}{3(n(n+1)-2)}=\\
\frac{4}{3(n+2)}.\\
\end{align}$$
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