Motor vehicles arrive at a toll gate according to a Poisson process with rate λ = 2 vehicles per minute. The vehicles arriving at the gate belong to classes 1,2 or 3 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 or $5 depending on the classes 1, 2, or 3 their vehicles belong to. Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the prob- ability that only $5 dollars are collected and all are collected during the first 5 minutes.

Motor vehicles arrive at a toll gate according to a Poisson process with rate λ = 2 vehicles per minute. The vehicles arriving at the gate belong to classes 1,2 or 3 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 or $5 depending on the classes 1, 2, or 3 their vehicles belong to. Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the prob- ability that only $5 dollars are collected and all are collected during the first 5 minutes.