6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye.
The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table.
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.6 | \(p\) | \(q\) | 0.05 |
It is given that \(\mathrm { E } ( X ) = 0.55\).
- Find the values of \(p\) and \(q\).
- Find \(\operatorname { Var } ( X )\).
Jim is practising for a competition and he repeatedly throws three darts at the board. - Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
- Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.