Problems below concern the model of Section 2.4 with a = ∞.

(a) For a separate moment of time, the odds are 3 to 1 in favor of gains. Which initial capital should company start with for the ruin probability to be less than 0.01?

(b) Show that q₁ ≥ q and explain, from a heuristic point of view, that it must be expected.

(c) Mark starts a risky business in which, on each day, he will make a profit of $500 with a probability of p = 0.9, but may suffer a loss of $500 with a probability of q = 0.1.
Mark takes a risk starting with zero initial capital, and hopes that he will not lose the first time and will be always able to cover losses in the future from the capital he will earn. Find the probability that Mark’s hope will come true. For which p (perhaps not equal to 0.9) is this probability larger then 0.95?