Question 39
Denote by $\langle x\rangle$ the fractional part of the real number $x$ (for instance, $\langle 3.2\rangle=0.2$ ). A positive integer $N$ is selected randomly from the set $\{1,2,3, \ldots, M\}$, with each integer having the same probability of being picked, and $\left\langle\frac{87}{303} N\right\rangle$ is calculated. This procedure is repeated $M$ times and the average value $A(M)$ is obtained. What is $\lim _{M \rightarrow \infty} A(M)$ ?