[Thrun] And again the solution follows directly from the state diagram over here. In the beginning we do know we're in state A and the chance of remaining in A is 0.5. This is the 0.5 over here. We can just read this off. For the next state we find ourselves to be with 0.5 chance to be in A and 0.5 chance to be in B. If we're in B, we transition with certainty to A. That's because of the 0.5. But if we're in A, we stay in A with a 0.5 chance. So you put this together. 0.5 probability being in A times 0.5 probability of remaining in A plus 0.5 probability to be in B times 1 probability to transition to A. That gives us 0.75. Following the same logic but now we're in A with 0.75 times a 0.5 probability of staying in A plus 0.25 in B, which is 1 minus 0.75, and the transition's uncertainty back to A as 1, we get 0.625. So now you should be able to take a Markov chain and compute by hand or write a piece of software the probabilities of future states. You will be able to predict something. That's really exciting.