A fair n-faced die (n1) is rolled repeatedly until a number less than n appears.

Question

A fair $n$-faced die ($n>1$) is rolled repeatedly until a number less than $n$ appears. If the mean number of tosses is $\dfrac{n}{9}$, find $n$.

Answer: 10

Accepted Answer

With $p=P(\text{success})=\dfrac{n-1}{n}$, mean of geometric distribution $=\dfrac1p=\dfrac{n}{n-1}$.

$$\frac{n}{n-1}=\frac{n}{9}\implies n-1=9\implies n=10$$

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