Question

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate α = 14 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.
(a) What is the probability that exactly ten arrive during the hour and all ten have no violations? (Enter your answer to five decimal places.)

Answer 1

According to Poisson, the probability that exactly 10 arrivals is
$$
\!f(k; \lambda)= \Pr(X{=}k)= \frac{(\lambda t)^k e^{-(\lambda t)}}{k!}
$$
Where \(t=1~hr,\lambda=14~cars~per~hr\) and \(k=10~cars\):
$$
\!f(k; \lambda)= \Pr(X{=}10)= \frac{(14)^{10} e^{-14}}{10!}
$$
And the probability of all 10 failing is
$$
0.5^{10}
$$
So the probability of 10 arrivals and all 10 failing is, because of independence:
$$
P=\frac{(14)^{10} e^{-14}}{10!}\times 0.5^{10}
$$
Make sense?

Answer 2

I should have said the probability of no violations is
$$
0.5^{10}
$$
But the answer comes out the same. Keep only 5 decimal places after rounding.