Rolling a 6
What is the expected number of rolls until a 6 is rolled on a fair six-sided die?
Answer
Using the geometric distribution, the number of rolls is geometrically distributed with success probability \(p=\frac{1}{6}\). Thus, the expected number of rolls is:
\[
E[X] = \frac{1}{p} = 6.
\]
Alternative (Recursion): Let \(E\) be the expected number of rolls. Then,
\[
E = \frac{1}{6}\cdot 1 + \frac{5}{6}\cdot (1+E).
\]
Solving this equation gives:
\[
E = \frac{1}{6} + \frac{5}{6} + \frac{5}{6}E \quad \Longrightarrow \quad E = 1 + \frac{5}{6}E,
\]
\[
E - \frac{5}{6}E = 1 \quad \Longrightarrow \quad \frac{1}{6}E = 1 \quad \Longrightarrow \quad E = 6.
\]