Freezesort

Since the vector $A=(\underline{8},\underline{3},\underline{2},4,5,6,7,\underline{1},9,10,11,12,13)$ has only the underlined elements out of order, we just need to choose and permutate subsets {8,1} and {3,2}. For both of them, the probability of getting the right order follows a geometric distribution $\text{Geo}\big(p=\frac{1}{2}\big)$ , so the expected number of steps to get one of them in order is $E_{\{8,1\}}=E_{\{3,2\}}=\frac{1}{p}=2$ . Evenutally, the expected number of steps to get the whole array sorted is $E= E_{\{8,1\}}+E_{\{3,2\}}=\boxed{4}$ .