A break room can hold max 20 people, if any more come by they have to wait outside You decide to count the number of people (they always come alone) passing by, in 10-minute intervals between 11.30 - 12.30. We assume that the number of people heading towards the break room in any 10-minute interval between 11.30 - 12.30 is a random variable, $N$. You repeatedly measure over the course of a few days, and conclude that approximately follows a Poisson distribution with $\lambda = 5$. Assume the break room is empty at 11:30.
a) What is the probability that 4 people, or fewer, enter the break room in the interval 11:30 - 11:40? Here - I have calculated like this: $$Px(X)=\frac{\lambda^x e^{-\lambda}}{x!}$$ $$Px(4) =\frac{5^4×e^{-5}}{4!}$$
b) Assume then that one day there are 16 people in the break room by 12:10. How probable are you to get company, of someone forced to wait outside, within the coming 20 minutes? I have done like this: $$P[X\ge 4] = 1 - P[X \le 4] = 1 - \left[\frac{5^0 × e^{-5}} {0!} +\frac{5^1 × e^{-5}}{1!} +...\right]$$
Have I done this correctly?
The random variable $N$ which follows a Poisson distribution with intensity parameter $5$, measures the number of people heading towards the break room every $10$ minutes interval However in part b of the question, we are required to find the probability that more than 4 people heads towards the break room in the interval $12:10$ to $12:30$ i.e. a $20$ min interval.
Follow this approach: Let $N_1, N_2$ be the random variables measuring the no of people heading in the $10$ min intervals: $12:10-12:20$, $12:20-12:30$. Thus, the total no of people heading towards the room is $N= N_1+N_2$ Now note that, $N_i$'s are iid (independent and identically distributed) random variables. And we have a property of poisson distribution that if $X$ and $Y$ follows $poi(\lambda)$ and $poi(\mu)$ respectively, $X+Y$ follows $poi(\lambda+\mu)$. The proof is quite simple, you can try it on your own.
Thus, $N$ follows $poi(5+5)$ or $poi(10)$. Now you can use the process you have used, except your intensity parameter would be $10$.
Also in part a we are required to find $4$ or fewer people entering the room, i.e. $P(N≤4)$ but you have calculated $P(N=4)$
Note: The independence of the random variables $N_1$ and $N_2$ is supported by the fact that the number of people heading toward the room in ANY 10 min interval is a $poi(5)$ random variable, which is an approximation made in the given question. In real world however, this independence may not hold true.