Accompanying code can be found on GitHub.
Let Xi be the random variable denoting the roll of a single die; let Yj = X1 + X2 + X3 be the random variable denoting the roll of three die to generate an ability score; let Z = max(Y1,Y2,Y3) be the random variable denoting the maximum of three trials to generate an ability score (using the “fun” method). All Xi are independent uniform IID; thus all Yj are independent uniform IID, and all Zk are independent uniform IID.
1. (a) Three Bernoulli trials
(b) Complement of a binomial distribution
(c) Each trait is a Bernoulli trial
P(Zi = 18, 1 ≤ i ≤ 6) = (P(Z = 18))6
(d) Get the set where all sums are ≤ 9, but at least one is equal to 9
P(Z = 9) = P(all sums ≤ 9) − P((all sums ≤ 9) ∧ (no sums = 9))
P(Zi = 9, 1 ≤ i ≤ 6) = (P(Z = 9))6
2. Let Xi denote the random variable representing the hitpoints (hp) of a goblin, and Yj denote the random variable representing the damage (dmg) of a fireball shot. Note that Xi ∪ Yj are mutually independent.
(a) Normal expected value
(b) We can enumerate the pmf by inspection
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = 0.25
P(Y = 2) = P(Y = 4) = 0.25 P(Y = 3) = 0.5
(c) We break up this question using a partition of Y .
P(slay all 6) =
(d) We can break down the event in question into a partition of threeevents (note that it is not possible that the surviving troll has ≤ 2 hp or that the firebolt did 4 dmg):
i. hp of surviving troll = 4, dmg = 3, all other trolls have hp ≤ 3
ii. hp of surviving troll = 4, dmg = 2, all other trolls have hp ≤ 2
iii. hp of surviving troll = 3, dmg = 2, all other trolls have hp ≤ 2
The probabilities of these events are easy to calculate.
Using Bayes’ rule, we can calculate the posterior pmf of X given that five trolls didn’t survive. Let W denote the event that the other five trolls died, and W = (i) ∪ (ii) ∪ (iii) ⇒ P(W) = P((i)) + P((ii)) + P((iii)) (union becomes addition since the events are disjoint). Then:
This in turn can be used to calculate the expected hp of the surviving troll:
(e) Let Zi denote the random variable denoting a roll of the 20-sided die (to decide whether Shedjam can hit Keene or not), Wj denote a roll of the 6-sided die (the Sword of Tuition’s damage), and Vk denote a roll of the 4-sided die (the Hammer of Tenure Denial’s damage).
E[dmg] = E[dmgSoT + dmgHTD]
= P(hitSoT)E[dmgSoT|hitSoT] + P(hitHTD)E[dmgHTD|hitHTD]
