This is a practice exercise on Poisson processes (not homework). Suppose that cars and trucks move between destinations A and B. Their movement can be described as two independent Poisson processes with respective rates 4 cars and 3 trucks per minute.
1) What is the probability that a person who started hitchhiking at 17:30 will first encounter a car?
2) What is the probability that the first three vehicles consist of two cars and one truck?
3) What is the probability that between 17:35-17:40 a person will encounter 3 cars and 2 trucks?
I know that the question is on composition of independent Poisson processes. However, I am not sure how I should approach solving the exercise, especially 1) and 2). Also, I am a bot confused when I should use joint probability and conditional probability for Poisson processes. In other words, when $P\{N(t)=1 | N_{1}(t)=1 \}$ vs $P\{N(t)=1,N_{1}(t)=1 \}$ where $N(t)=N_{1}(t)=1 + N_{2}(t)=1$.
Vehicles are arriving in two independent Poisson processes at rates of 4 cars and 3 trucks per minute.
Then the vehicles are arriving in a Poisson process with a rate of seven vehicles per minute, where each arrival has an independent probability of 4/7 for being a car, or of 3/7 for being a truck.
(1) Is a Bernoulli Trial of the given success rate, and (2) involves a Binomially Distribution of three Bernoulli trials of the given success rate.
The joint probability mass function for two independent random variables is the product of the probability mass function for each.