1. She rolls the dice twenty times and records the number of times that it lands on a 6.  

Find the probability that

(i) the dice lands on a ‘6’ four times

(ii) the dice lands on a ‘6’ four times, but all of those sixes occur within the first  $k$ rolls (where $4\leqk\leq20$).

Your answer for (ii) should be given in terms of $k$, in the form

            $(\begin{matrix}a\\b\end{matrix}){(\frac{p}{16})}^{q}{(\frac{r}{16})}^{s}$

where  $(\begin{matrix}a\\b\end{matrix})=\frac{a!}{b!(a-b)!}$ is a binomial coefficient, and $a,b,p,q,rands$ are constants to be found.