Suppose that the number of trucks arriving at a depot per day is a random variable having a Poisson distribution with $\lambda = 28.8$. What is the probability that the time between two arrivals is at least 1 hour?
This seems similar to part (b) of this question.
My attempt:
This is the same as the probability of the first arrival (after 9AM say) being at least 1 hour later.
I thought that it would be $$\int_{1/24}^\infty 28.8 e^{-28.8t}dt = e^{-1.2} \approx .3012,$$ but the answer in the back of the book is $.1827$ instead.
"Another" attempt:
An average of $28.8/24 = 1.2$ trucks arrive per hour. Let $P(x;\lambda)$ be the probability distribution of a Poisson random variable. Then we want $$P(0; 1.2) = \frac{1.2^0e^{-1.2}}{0!} = e^{-1.2} \approx .3012.$$
As an aside, I'll mention that I think this exercise is flawed because presumably, the rate at which the trucks arrive would be different in the middle of the day than in the middle of the night.