Question

Three out of every four trucks on the road are followed by a car, while only one out of every five cars on the road is followed by a truck. In the long run, what proportion of vehicles on the road are trucks?

Answer 1

Let Cn be the number of cars among the first n vehicles on the road. We can then relate the number of cars among the first n vehicles to the number of cars among vehicles 2 through n+ 1 by counting the number among following vehicles. Thus, we would directly get the relation

Cn+1 - C1 = Tn(3/4) + Cn(4/5),

where also,

Cn + Tn = n.

Combining these two equations, we get

Cn+1 − C1 = (n − Cn)(3/4) + Cn(4/5) = n(3/4) + Cn(1/20) ............. (i)

We should expect that, as n → ∞, Cn/n → C.

Dividing by n in (i) and letting n → ∞, we get

C = 3/4 + C(1/20) or C(19/20) = 3/4,

so that C = 15/19