Let X be a Markov chain with state space {a,b,c,d} and transition probabili- ties given by P 0.1 0.3 0.6 0.0 0.0 0.5 0.2 0.5 0.3 0.0 0.5 0.0 0.0 0.0 1.0 0.0 Each time the chain is in state a, a profit of $20 is made; each visit to state b yields a $5 profit; each visit to state c yields $15 profit; and each visit to state d costs $10. Find the following: (a) E[(X5)|X3 = c, X4 = d]. (b) E[ (X) Xo = b]. (c) E[(X)|Xo = b].