Question

In D&D, a short sword usually does 1d6 damage. But you have a special short sword named Jabby where you roll 2d6, and the damage is the max of those two.

Find the expected value of both of these random variables

Don’t do the sum of 2d6, do the maximum

Write a sentence interpreting what the expected value of the max of 2d6 tells us about Jabby

Answer 1

Expected value when we roll one 6 face die:

Probability mass function,

X	1	2	3	4	5	6
P(X)	1/6	1/6	1/6	1/6	1/6	1/6

= 1 * (1/6) + 2 * (2/6) + ------- + 6*(1/6)

= 21 / 6

= 3.5

Expected value when we roll two 6 face dice:   

X is maximum of two die. Probability mass function of X is :

X	Sample points	Probability
1	(1,1)	1/36
2	(1,2), (2,1), (2,2)	3/36
3	(1,3), (2,3), (3,1), (3,2), (3,3)	5/36
4	(1,4), (2,4), (3,4), (4,1), (4,2), (4,3), (4,4)	7/36
5	(1,5), (2,5), (3,5), (4,5), (5,1), (5,2), (5,3), (5,4), (5,5)	9/36
6	(1,6), (2,6), (3,6), (4,6), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5 ), (6,6)	11/36

= 1 *(1/36) + ------------ + 6 * (11/36)

= 161/36

= 4.4722

Average of mximum of two dice is 4.4722