6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\).
- Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\).
\includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328} - Find \(\mathrm { P } ( X = 4 )\).
- Find \(\mathrm { P } ( X > 4 )\).
- Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).