Problem 3. Find the most probable number of hits in the ring in five throws, if the probability of hitting the ring with the ball in one throw is $p=0.6$.
Solution. We have $n=5 ; p=0.6 ; q=0.4$. For the number $k_{0}$, we obtain the estimate: $5 \cdot 0.6-0.4 \leqslant k_{0} \leqslant 5 \cdot 0.6+0.6$, i.e., $2.6 \leqslant k_{0} \leqslant 3.6$. Since $k_{0}$ is an integer, then $k_{0}=3$.

Direct calculations of $p_{5}(k)$ lead to the values: $p_{5}(0)=$ $=0.01024 ; p_{5}(1)=0.0768 ; \quad p_{5}(2)=0.2304 ; p_{5}(3)=0.3456 ; p_{5}(4)=$ $=0.2592 ; p_{5}(5)=0.07776$. The largest of the numbers $p_{5}(k)-$ is $p_{5}(3)=$ $=0.3456$; it corresponds to the value $k_{0}=3$.

Answer. 3 .