Question

A fair six-sided green die is rolled resulting in a number between 1 and 6. Then that number of red dice are rolled. Find the expected value and variance of the sum of the red dice.

Answer 1

Intuitively we would expect the sum of a single die to be the average of the possible outcomes, ie:

S=(1+2+3+4+5+6)/6=3.5

And so we would predict the sum of a two die to be twice that of one die, i.e. we would predict the expected value to be 7

Expected Value of 2 dice is 7

First, the definition of expected value is E(X)=∑x⋅p(x)E(X)=∑x⋅p(x). For a die roll, this is 1/16+2⋅1/6+3⋅1/6+4⋅1/6+5⋅1/6+6⋅1/6=21/6

Second, the expected value is linear, meaning that the expected value of the sum of random variables is the sum of their expected values. This means that if you roll n dice, the expected value of the sum of the faces is n times the expected value of a single face, or n*1/6

Expected values of the expected value of the red die.

21/6 + 2*21/6 + 3*21/6+ 4*21/6 + 5*21/6 + 6*21/6 = 15*(21/36) = 105/12 = 8.75