Two motorcycles among four vehicles

Question

At a checkpoint, cars pass as a Poisson process at rate $15$ per hour and motorcycles independently at rate $5$ per hour. In a $10$ -minute period, exactly $4$ vehicles passed in total.

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

Accepted Answer

Given the total number of vehicles, each one was independently a car or a motorcycle, with probabilities set by the rate shares, so the motorcycle-count is binomial. The chance any one vehicle is a motorcycle is p = \frac{\lambda{	ext{moto}}}{\lambda{	ext{car}} + \lambda{	ext{moto}}} = \frac{5}{15 + 5} = 0.25. The number of motorcycles among the 4 is Binomial(4, 0.25). The probability of exactly 2: P(2) = \binom{4}{2}(0.25)^2(0.75)^2 = 6 	imes 0.0625 	imes 0.5625 = 0.210938 \approx 0.211. About a 21\% chance. The 10-minute window and the absolute rates never entered once the total was fi…

Two motorcycles among four vehicles

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