Please help on this Applied Stochastic Processes/ Statistic question

n Sn = So+ > Xi (n > 1) i=1 be a simple random walk starting in the random variable So. That is, X1, X2, . .. is a sequence of i.i.d. random variables independent of So such that P[X; = +1] =p and P[X, = -1] =1 -p for all i > 1 and some pe (0, 1) (a) Show that E[Sn+1 \\ Sn] = Sn + 2p -1 for all n > 0. (b) Use the result in (a) to determine for which values pe (0, 1) it holds that (i) E[Sn+1 \\ Sn] = Sn, (ii) E[Sn+1 | Sn] > Sn, (iii) E[Sn+1 | Sn] 0. (c) Suppose that (Sn)n=0,1,2,... denotes the profit and loss from $1 bets of a gambler with initial capital So who is repeatedly playing a game with probability of winning and losing her stake of $1 given by p and 1 -p, respectively. What are the interpretations of the results in (b) in terms of the probability p of winning