Problems

A set of \(n\) dice is rolled repeatedly. For each die the probability of showing a six is \(p\). Show that the probability that the first of the dice to show a six does so on the \(r\)th roll is $$q^{n r } ( q^{-n} - 1 )$$ where \(q = 1 - p\). Determine, and simplify, an expression for the probability generating function for this distribution, in terms of \(q\) and \(n\). The first of the dice to show a six does so on the \(R\)th roll. Find the expected value of \(R\) and show that, in the case \(n = 2\), \(p=1/6\), this value is \(36/11\). Show that the probability that the last of the dice to show a six does so on the \(r\)th roll is \[ \big(1-q^r\big)^n-\big(1-q^{r-1}\big)^n. \] Find, for the case \(n = 2\), the probability generating function. The last of the dice to show a six does so on the \(S\)th roll. Find the expected value of \(S\) and evaluate this when \(p=1/6\).