Question 10.7: Show that BPP ⊆ PSPACE.

If M is a probabilistic TM that runs in polynomial time, we can modify M so that it makes exactly n^{r} coin tosses on each branch of its computation, for some constant r. Thus, the problem of determining the probability that M accepts its input string reduces to counting how many branches are accepting and comparing this number with \frac{2}{3}2^{(n^{r} )}. This count can be performed by using polynomial space.