Two of five ER patients are children

Question

In an emergency room, adult patients arrive as a Poisson process at rate $6$ per hour and child patients independently at rate $4$ per hour. In a particular hour, exactly $5$ patients arrived in total.

Given that, what is the probability that exactly $2$ of them are children?

Accepted Answer

Here is the key idea: once you know the total number of arrivals from two merged Poisson streams, each arrival was independently a child or an adult, with fixed probabilities set by the rates. So conditioned on the total, the number of children follows a binomial distribution. The chance any given patient is a child is that stream's rate share: p = \frac{\lambda{	ext{child}}}{\lambda{	ext{child}} + \lambda{	ext{adult}}} = \frac{4}{4 + 6} = 0.4. So the number of children among the 5 patients is Binomial(5, 0.4). The probability of exactly 2: P(2) = \binom{5}{2}(0.4)^2(0.6)^3 = 10 	imes 0.1…

Two of five ER patients are children

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