Car arrive at a toll booth according to the Poisson process at a rate of 3 arrivals per minute.
a) What is the probability that the third car arrives within 3 minutes of the first car?
b) Of the cars arriving at the booth, it is known that over the long run 60% are Japanese imports. What is the probability that in a given 10 minute interval, 15 cars arrive at the booth and 10 of these are Japanese imports?
For a) I'm thinking that we need to find the P(Time two more cars arrive after first car < 3), but I'm not sure how to set the any formulas up.
For b) We assume independence so we can use the multiplication rule for P(10 cars out of 15 are Japanese) and P(15 cars arrive in 10 minutes). Again, not sure how to use the Poisson process to set up the second probability.
Best Answer
For $\lambda=3$/minute, the $\mathsf P(N=k\mid T=t) = \dfrac{(3t)^k\mathsf e^{-3t}}{k!}$
Hint: You don't care what time the first car arrives. You want the probability that in the following three minutes at least two more cars arrive.
$$\mathsf P(N \geq 2\mid T=3) = 1- \sum_{k=0}^1 \mathsf P(N=k\mid T=3)$$
For a given number of cars that arrive, $N$, then the count of imports is the count of successes in $N$ Bernoulli trials with success rate $0.60$. Under this condition, what kind of distribution is this?
$$\mathsf P(N=15, N_\text{import}=10 \mid T=10) ~=~ \mathsf P(N=15\mid T=10)\cdot\mathsf P(N_\text{import}=10 \mid N=15)$$