To solve this, assume that traffic accidents is a Poisson point process (https://en.wikipedia.org/wiki/Poisson_point_process)
Random time T between 2 accidents has an exponential distribution with a parameter λ and a cumulative distribution function P(T < t) = 1 - e-λt (let measure time in hours)
Random number X of accidents during 1 hour has a Poisson distribution with the same parameter λ with the expected value E(X) = λ.
Given that the average number of accidents is 4 per hour, so λ = 4.
P(the next accident will take place more than 30 minutes (.5 hour) from the previous one) = P(T > 0.5) =
= 1 - P(T ≤ 0.5) = 1 - (1 - e-4(0.5)) = e-2
P(the next accident will take place between 15 and 30 minutes (0.25 and 0.5 hour) from the previous one) =
= P(0.25 < T < 0.5) = P(T < 0.5) - P(T < 0.25) = (1-e-4(0.5)) - (1-e-4(0.25)) = e-1 - e-2
