I got an exercise I don't know how to solve. Maybe you can help?
Suppose that there are three fashion brands, which we call $A$, $B$ and $C$. Customers tend to stick to the same brand. Those who choose type $A$ choose it the next time around with probability $0.8$; those who choose type $B$ choose it next time with probability $0.6$. The probabilities for types $C$ is given by $0.7$, respectively. When customers do change a brand, they choose one of the other two equally probably.
Compute the probabilities: $\mathbb P\lbrace X_4 = C,X_3 = C,X_2 = A,X_1 = A | X_0 = A\rbrace$
So ... is it correct to say, just for the understanding, that I have to calculate the probability, that customer $4$ chooses $C$, and $3$ $C$, and $2$ $A$ and $1$ $A$ if $0$ chooses $A$ beforehand?
If so, how ... how would I have to calculate it?