Poisson Distribution vs Gamma distribution

Question

Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 minutes. Motorcycles arrive about once every 30 minutes. The question : Find the probability that the seventh motorcycle arrives within 45 minutes of the third motorcycle? The answer is:

$$ P(S_7-S_3<45)=P(G<45)=pgamma(45,4,1/30), where G~Gamma(4,1/30)=0.066 $$ My question is why the solution is not a poisson solution? $$ (_{45} = 4|_{+45} = 7) = (_{45} = 4|_ = 3)=(_{45} = 4)=0.047? $$

Answer 1

Your Poisson calculation would work if stated correctly. $N_{t}\sim\text{Pois}(\frac{t}{30})$ here

You want $P(N_{45} \ge 7-3) = 1- P(N_{45} \le 3)$: there may be $4$ or more motorcycles in the $45$ minutes (your $0.047$ is for exactly $4$)

so you can find it in R with

1 - ppois(3, 45/30)
# 0.06564245

the same as pgamma(45, 4, 1/30)

Answer 2

The Gamma distribution is the waiting time distribution time between $n$ arrivals in a Poisson process.