Probability of an event occuring in a subinterval, given that some number of events ocurred on the interval

Question

I've just started delving into probability, and I was presented this exercise:

Patients get to a hospital in a mean rate of $5$ patients per hour, following a Poisson process. Given that $13$ patients arrived between $12$PM and $3$PM, what is the probability that $10$ patients arrived between $12$PM and $2$PM.

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How can I go about solving this?

Answer 1

Write $N_{(a,b]}$ for the number of patients that arrive between time $a$ and $b$, where $a$ and $b$ are hours measured from noon. (Adjust the notation $N_{(a,b]}$ to fit your endpoint convention.) You are looking for $$ P(N_{(0,2]}=10\mid N_{(0,3]} = 13)=\frac{P(N_{(0,2]}=10, N_{(0,3]} = 13)}{P( N_{(0,3]} = 13)} $$ The event in the numerator can be written as the event $\{N_{(0,2]}=10, N_{(2,3]} = 3\}$, which is an intersection of independent events.

Answer 2

Hint

If $X\sim Pois(\lambda _1)$ and $Y\sim Pois(\lambda _2)$ are independents, one can prove that $X+Y\sim Pois(\lambda _1+\lambda _2)$.