8.
The discrete random variable \(R\) takes even integer values from 2 to \(2 n\) inclusive.
The probability distribution of \(R\) is given by
$$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$
where \(k\) is a constant.
- Show that \(k = n ( n + 1 )\)
When \(n = 20\)
- find the exact value of \(\mathrm { P } ( 16 \leqslant R < 26 )\)
When \(n = 20\), a random value \(g\) of \(R\) is taken and the quadratic equation in \(x\)
$$x ^ { 2 } + g x + 3 g = 5$$
is formed.
- Find the exact probability that the equation has no real roots.