Question

A die with six sides is rolled repeatedly and summed. The expected number of rolls until the sum is a multiple of 6 is

Answer 1

There are two things you need to know about in order to compute this expected value. First, the definition of expected value is E(X)=∑x⋅p(x)E(X)=∑x⋅p(x). For a die roll, this is 1⋅1/6+2⋅1/6+3⋅1/6+4⋅1/6+5⋅1/6+6⋅1/6=21/6.

Second, 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 nn dice, the expected value of the sum of the faces is nn times the expected value of a single face, or 21/6n. So the answer you found was correct.