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 \(\frac { 2 } { 3 }\).
Find
- \(\mathrm { P } ( X = 4 )\),
- \(\mathrm { P } ( X > 4 )\),
- the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).