HCI Sampling Tutorial Q9

A circular card, with a pointer pivoted at the centre is divided into \(5\) unequal sectors numbered ‘\(1\)’, ‘\(2\)’, ‘\(3\)’, ‘\(4\)’ and ‘\(5\)’. The pointer is spun, and the score will be the number at which the pointer stopped at. The probability of scoring a ‘\(5\)’ is \(1-q\). The pointer is spun \(10\) times independently and the number ‘\(5\)’’s obtained is denoted by \(Y\). Given that \(\mathrm{Var}\left( Y \right)={{\left[ \mathrm{E}\left( Y \right) \right]}^{2}}\), show that \(q=\frac{10}{11}\).

Supposed that \(50\) people are invited to spin the pointer \(10\) times each, find the probability that the mean number of times they obtain a ‘\(5\)’ exceeds \(1\).