A random variable W has a geometric distribution, $W \sim Geo(p)$, counting the number of trials up to and including the first success. Given that $Var(W) = 30$, find the value of p.

For a geometric distribution on 1, 2, 3, ... the variance is $\operatorname{Var}(W)=\dfrac{1-p}{p^{2}}$.

Setting this to 30 gives $1-p=30p^{2}$, so $30p^{2}+p-1=0$.

Factorising, $(6p-1)(5p+1)=0$, so $p=\dfrac{1}{6}$ or $p=-\dfrac{1}{5}$.

A probability cannot be negative, so $p=\dfrac{1}{6}=0.1667$.