NYJC Class Quiz

An ordinary unbiased die is thrown until the first six is obtained. The random variable \(T\) denotes the number of throws in order to obtain the first six.

1. Write down \(\mathrm{P}(T=r)\) in terms of \(r\) where \(r=1\,,2\,,3\,,.....\)[1]
2. By summing the series \(1+x+{{x}^{2}}+{{x}^{3}}+...\,\,\left( \left| x \right|<1 \right)\) and using differentiation, show that
   \(\sum\limits_{r=1}^{\infty }{r{{x}^{r-1}}=\frac{1}{{{(1-x)}^{2}}}}\) where \(\left| x \right|<1\).[2]
3. Hence deduce the exact value of \(\mathrm{E}(T)\).[2]