Q. 9.3 - A First Course in Probability

Question

Four out of every five trucks on the road are followed by a car, while one out of every six cars is followed by a truck. What proportion of vehicles on the road are trucks?

Step-by-Step Solution

Answer

$\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$29$}\right.$ percent of vehicles on the road are trucks.

Step 1: Given Information

We need to find the proportion of vehicles on the road are trucks.

Step 2: Simplify

The Markov chain has two states. Let state$1$ be a car and let state $2$ be a truck. From the text we have that the transition matrix is

$P=\left[\begin{array}{cc}5/6& 1/6\\ 4/5& 1/5\end{array}\right]$

Finding the stationary distribution. Solving $\mathrm{\pi }=\mathrm{\pi P}$ which is

$$\begin{gathered} {\mathrm{\pi }}_1=\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6{\mathrm{\pi }}_1$}\right.+\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5{\mathrm{\pi }}_2$}\right. \\ {\mathrm{\pi }}_2=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6{\mathrm{\pi }}_1$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5{\mathrm{\pi }}_2$}\right. \end{gathered}$$

Marking that it has to be ${\mathrm{\pi }}_1+{\mathrm{\pi }}_2=1$,

${\mathrm{\pi }}_1=\frac{24}{29},{\mathrm{\pi }}_2=\frac{5}{29}$

which implies that $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$29$}\right.$ percent of vehicles are trucks.