Each time a shopper purchases a tube of toothpaste, he chooses either brand A or brand B. Suppose that for each purpose after the first, the probability is 1/3 that he will choose the same brand that he chose on his preceding purchase and the probability is 2/3 that he will switch brands. If he is equally likely to choose either brand A or brand B on his first purchase, what is the probability that both his first and second purchases will be brand A and both his third and fourth purchases will be brand B?

http://www.bilkent.edu.tr/~math250/Solution%20set2.pdf