11 The discrete random variable \(X\) takes the values \(0,1,2,3,4\) and 5 with probabilities given by the formula
$$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$
- Find the value of \(k\).
In one half-term Ben attends school on 40 days. The probability distribution above is used to model \(X\), the number of lessons per day in which Ben receives a gold star for excellent work.
- Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
- Find the expected number of days in the half-term on which Ben receives no gold stars.