Question

(1 point) You're playing Dungeons and Dragons. Each round, you make an attack. If the attack hits, you deal a certain amount of damage (otherwise, you deal zero damage for that round).

First, you roll a twenty-sided die to see you hit. (Assume the die is fair etc.) If you roll 11 or greater, you make a hit, then roll two six-sided dice; the damage you deal is the sum of these dice.

1. What is the expected value of the amount of damage you deal in a round assuming you hit?

2. What is the expected value of the amount of damage you deal in a round (with no assumptions)?

3. What is the expected value of the amount of damage you deal in 20 rounds?

(1 point) You're still playing Dungeons and Dragons; each round, you make an attack, but with a different weapon. This time, the roll of the twenty-sided die needs to be 13 or greater, and the roll for the damage is a single ten-sided die (with numbers from 1 to 10).

4. In addition, if you roll a 20 on the first die, you make a "critical hit", and whatever damage you deal is doubled. For example, you roll a 20 on the twenty-sided die, then a 3 on the ten-sided die, you deal 3⋅2=63⋅2=6 damage.
What is the expected amount of damage you deal in a single round?

Answer 1

Let the event of rolling an 11 or more on a 26-sided die be A.

P(A) = 16/26

1.

Probability distribution of sum on two dice:

Expected value of the amount of damage you deal in a round assuming you hit, E₁ = Expected value of the sum on the two dice

= ∑x*P(X=x)

= 252/36

= 7

2.

Expected value of the amount of damage you deal in a round (with no assumptions), E₂ = P(A)*E₁

= (16/26)*7

= 4.3077

3.

For this case,

P(A) = 14/26

E₁ = Expected value of sum on a ten-sided die

= (1+2+....+10)/10

= 5.5

E₂ = 5.5*14/26

= 2.96

In 20 rounds,

Expected damage = 20*2.96

= 59.23

4.

In this case,

E₂ = (13/26)*5.5 + (1/26)*11 [Because for '20' on the first round you get double of what you have dealt]

= 2.75 + 0.423

= 3.173

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