Dice Roll

Choose a number from 1 to 6. Roll a fair, 6-sided die numbered 1 to 6. A roll less than or equal to the chosen number is a hit, otherwise it's a miss. What is the expected number of rolls needed to hit a number of times equal to the chosen number?

Answer

Let the chosen number be \(x\) (\(1\le x\le6\)). A roll is a hit if it is \(\le x\), which happens with probability

\[ p=\frac{x}{6}. \]

Since we need \(x\) hits, the expected number of rolls is given by the negative binomial formula:

\[ \text{Expected rolls}=\frac{x}{p}=\frac{x}{\frac{x}{6}}=6. \]

Thus, the expected number of rolls is 6.

Notes and comments

Comment 1: We, of course, could have also done it as the sum of geometrically distributed RV's.

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