n-sided die

Let $X_{i}$ be a random variable, defined as follows: 1 if all dice rolls up to and including the i'th dice roll are not side $S_{i}$ . Else it takes the value 0. Now we see the problem can be phrased in terms of calculating the following expected value problem: find $E[\sum_{i=0}^{\infty} X_{i}]$ . We know from LOE that $E[\sum_{i=0}^{\infty} X_{i}] = \sum_{i=0}^{\infty}E[X_{i}]$ .

Now let's calculate $E[X_{i}]$ . From the definition, it equals $1*P(X_{i} = 1) + 0*P(X_{i} = 0)$ . The second term vanishes and we need only find the probability that $X_{i} = 1$ . For this to be true, all dice rolls up to and including the i'th roll are not $S_{i}$ . Thus we have a probability of $(19/20)^i$ that $X_{i} = 1$ .

Now we use the geometric series to see that $\sum_{i=0}^{\infty}E[X_{i}] = \sum_{i=0}^{\infty} (19/20)^i = 1/(1-(19/20)) = 20$ .