In this lab, we’ll practice using the fundamental definitions of probability theory to perform some probabilistic computations.
Problem: Roleplaying
In tabletop games such as Dungeons and Dragons, players roll a variety of polyhedral die each with a certain number of sides. To express the number of such dice rolled, we use the notation \(xdy\) to refer to rolling \(x\) dice with \(y\) sides. For example \(1d6\) refers to rolling one six-sided die (with values 1–6). In contrast, \(2d8\) means rolling two eight-sided dice (with values 1–8).
Give combinatorial descriptions for each of the following values:
The probability of succeeding at a medium ability check in Dungeons and Dragons (with no additional modifiers). To succeed at a medium ability check, the player rolls \(1d20\) and succeeds if they get a 15 or higher.
A player has advantage if the ability check is made under circumstances that are favorable to the player. When a player has advantage, they roll \(2d20\) and take the higher of the two rolls. What is the probability that the player succeeds a medium ability check with advantage?
(Hint: you might find it easier to reason about the situations in which a player fails. The probability of success, by the axioms of probability, is one minus the probability of failure.)
A player has disadvantage if the ability check is made under circumstances unfavorable to the player. When a player has disadvantage, in contrast, they roll \(2d20\) and take the lower of the two rolls. What is the probability that the player succeeds a medium ability check with disadvantage?
(Hint: consider what rolls lead to player success in this scenario.)
Consider a level 6 Disintegrate spell. To resist the spell, the victim rolls a single \(d20\) and then adds their dexterity modifier \(d\) to the roll. This is compared to the modified spellcasting ability of the caster, calculated as follows:
\[ 8 + \text{spellcasting profiency bonus} + \text{intelligence modifier}. \]
Let \(s\) be the spellcasting bonus and \(i\) be the intelligence modifier. The victim resists the spell if their modified roll is higher than the modified spellcasting ability of the caster. What is the probability that the victim resists the spell?
If a Disintegrate spell is not resisted, the victim takes \(10d6 + 40\) damage. If this damage reduces the victim to \(0\) hit points or lower, the target is disintegrated—they cannot be resurrected except through a True Resurrection or Wish spell! Suppose that the dexterity modifier of the victim is \(1\), the spellcasting proficiency bonus of the caster is \(4\), the caster’s intelligence modifier is \(2\), and the victim has \(100\) hit points. What is the probability that the Distintegrate spell disintegrates the victim?
(Hint: with concrete numbers, you can compute (a) the total number of outcomes possible from the \(10d6\) roll and (b) the total number of ways that the damage roll exceeds the victim’s health.)