
Markov chain
Suppose three out of every four trucks on the road are followed by a car, while only one out of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?
I form a markov chain as followed:
$\displaystyle \[ \left( \begin{array}{ccc}
0.25 & 0.75 \\
0.20 & 0.80
\end{array} \right)\]$
Then find the limiting probabilities $\displaystyle \pi_{track}$ and I got 0.2105.
Am I right?

Yup, thats what i got, since its irreducible finite markov chain there is a stationary distribution, which is the proportion of time the chain spends in each state;
If 0 is the event of a truck, 1 is a car (as outlined in your transition matrix) then
$\displaystyle \pi_0=.25\pi_0+.2\pi_1$
$\displaystyle \pi_1=.75\pi_0+.8\pi_1 \to \pi_1=3.75\pi_0 $
And $\displaystyle \pi_0+\pi_1=1$
So solving,
$\displaystyle \pi_0=\frac{1}{1+3.75}=0.2105$