12. For a Markov chain {Xn,n 0} with transition probabilities Pi,j , consider the conditional probability that Xn = m given that the chain started at time 0 in state i and has not yet entered state r by time n, where r is a specified state not equal to either i or m. We are interested in whether this conditional probability is equal to the n stage transition probability of a Markov chain whose state space does not include state r and whose transition probabilities are Qi,j = Pi,j 1 − Pi,r

, i,j = r Either prove the equality P{Xn = m|X0 = i,Xk = r,k = 1,...,n} = Qn i,m or construct a counterexample.