Let’s say in a certain game, a player has two such attributes: “strength” (let’s call it S) and “critical chance” (let’s call it C). Critical chance is a number between 0 and 100. It gives the percentage chance that a hit will be a “critical hit.” (so the probability of a critical hit is C 100 ). A regular hit does S amount of damage; but a critical hit does M times the damage (M is a constant). The average damage dealt per hit is the probability of a critical hit times the amount of damage dealt by that hit, plus the probability of a regular hit times the amount of damage that it (a regular hit) deals. Let’s say our player has a total of 100 points that he/she can assign to either strength or critical chance, or both (so for example, C = 30, S = 70 is one option).

(a) What is the average damage if all points are put into strength?

(b) What is the average damage if all points are put into critical chance?

(c) If the “damage multiplier” M is 3, how should the 100 points be allocated in order to maximize the average damage?

(d) What if M = 1.5?