7 The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .
- Show that \(15 p ^ { 2 } + 4 p - 4 = 0\) and hence find the value of \(p\).
- Find \(\mathrm { P } ( X = 5 )\).
- Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\).