Dice Multiple
You repeatedly roll a fair 6-sided die until the total sum of the rolled numbers is a multiple of 6. Determine the expected number of rolls required.
Answer
Let \(E\) be the expected number of rolls. Observe that, from any current total (modulo 6), each new roll has a \(\tfrac{1}{6}\) chance of making the sum a multiple of 6 (thus ending the process) and a \(\tfrac{5}{6}\) chance of leaving it not divisible by 6 (so we “start over” in the same sense). Hence the recursion
\[
E \;=\; 1 + \tfrac{5}{6}\,E.
\]
Solving yields
\[
E - \tfrac{5}{6}E \;=\; 1
\quad\Longrightarrow\quad
\tfrac{1}{6}E \;=\; 1
\quad\Longrightarrow\quad
E \;=\; 6.
\]