# Expected number of rolls needed to get four distinct results

Suppose I have a fair, six-sided die. What is the expected number of rolls it would take me to get four distinct outcomes?

Mean number of rolls until the first: $$\mathbf{E}X_1 = 1$$
Mean number of rolls until the second: $$\mathbf{E}X_2 = \frac{1}{\frac{5}{6}} = \frac{6}{5}$$