Each vehicle arriving at a toll booth is either a car or a truck. Cars arrive as a Poisson process with rate = 3 per minute. Independently of the cars, trucks arrive as a Poisson process with rate = 1 per minute.

(a) What is the probability that exactly 5 cars have arrived by the time the second truck arrives?

(b) Given that in the time interval of [0, 20] there are in total 80 vehicles arriving, what is the conditional probability that there are precisely 40 vehicles arriving in the time interval [0, 10] and precisely 40 cars arriving in the time interval [10, 20]?

(c) Each car has to pay a toll of 3 dollars and each truck has to pay a toll of 5 dollars. What is the probability that the total amount of dollars collected in the time interval [0, 1] is strictly less than 7?