The problem is like this: You are on the origin of a number axis, each time, you can move forward, backward or stay with probability P_f, P_b, P_s = 1 - P_f + P_s respectively, i.e., if you are on x now, the next step you can move to x + 1 with probability of P_f and etc. And the question is: after n steps, what's the mathematical expectation of the maximal number you have reached.

Here's an example when n = 2, P_f = 0.25, P_b = 0.25, P_s = 0.5: ~~~~~ maximal number: 1 fs: 0.25*0.5 = 0.125 sf: 0.5*0.25 = 0.125 fb: 0.25*0.25 = 0.0625 (since you have arrived at number 1, even if you go back, you will have maximal number 1) ~~~~~ ~~~~~ maximal number: 2 ff: 0.25*0.25 = 0.0625 ~~~~~ since you are on the origin, all other path will have maximal number equal to 0

Thus the expectation is: 1*(0.125+0.125+0.0625)+2*0.0625=0.4375

There is a dp solution on the problem, but I just can't quite figure out why, perhaps you can just think it over first before I give the answer.