A repair protein by the name of helicase moves along a strand of DNA in a random walk driven by water molecules. If we assume an upwards movement with probability \(p=0.51\), and then down with probability \(q=0.49, \ldots\)

(a) what are the possible positions relative to the starting point after 10 steps;

(b) what is the probability of having moved at least 4 units up in 10 steps;

(c) what is the expected number of units moved upwards after 10000 steps;

(d) what is the variance in the number of units moved upwards after 10000 steps;

(e) what is the probability of having moved at least 100 units upwards after 10000 steps?