149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played.
Solution. It is known that if the product of the number of trials $\boldsymbol{n}$ and the probability $p$ of the event occurring in one trial is an integer, then the most probable number is

$$
k_{0}=n p .
$$

In the problem at hand, the number of trials $n$ is equal to the number of games played $2 N$; the probability of the event occurring is equal to the probability of winning in one game, i.e., $p=1 / 2$ (by the condition that the opponents are of equal strength).

Since the product $n p=2 N \cdot 1 / 2=N$ is an integer, the sought most probable number $k_{0}$ of games won is $N$.