Question

There are many different types of dice. Their principal features are that the faces are numbered sequentially beginning with 1, and that each face is equally likely to come up. In the second edition of the Advanced Dungeons & Dragons role-playing game, a broadsword deals damage equal to the roll of two four-sided dice (tetrahedrons). The battleaxe deals damage equal to the roll of one eight-sided die (octohedron). From a probability point of view, which is better and why?

Answer 1

The probability that any particular face will come up in a 4 sided die

= 1/4

The probability that any particular face will come up in an 8-sided die

= 1/8

The expected value of the outcome of a 4-sided die is

= 1 * 1/4 + 2 * 1/4 + 3 * 1/4 + 4 * 1/4

= 1/4 + 1/2 + 3/4 + 1

= 2.5

The expected value of the outcome of two 4-sided die is

= 2* 2.5 = 5

The expected value of the outcome of one 8-sided die is

= 1 * 1/8 + 2 * 1/8 + 3 * 1/8 + 4 * 1/8 + 5 * 1/8 + 6 * 1/8 +

7 * 1/8 + 8 * 1/8

= 1/8 + 2/8 + 3/8 + 4/8 + 5/8 + 6/8 + 7/8 + 8/8

= 36/8

= 4.5

As we can see, the expected value for two 4-sided die is greater than one 8-sided die.

Hence, for greater damage, from a probability point of view, it is better to use the broadsword than the battleaxe because the broadsword has a higher expected damage since it used two 4-sided die.

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