A hair shop, people arrives at the rate $1$ person/hour, and it spend $0.5$ hour to completely cut the hair. Find the probability to see $5$ peoples in the hair shop, including the person who are cutting?
Update: The ultilization $\rho=\frac{1}{2}$
The pro. has 5 people in hair shop is $P[M=5]=\sum_{k=1}^{5}(1-\rho)\rho^k=0.484375$
Is it correct? I am not sure about the range of $k$ in the sum
Hint: The probability to see exact 5 people in the system is
$$P_5=\rho^5\cdot (1-\rho) $$.
The sigma sign is not needed here. Your are right that $\rho=\frac{\lambda}{\mu}=\frac{1}{2}$.