2020 TJC P2 Q10

Anand and Charlie decide to do some exercise and have fun at the same time. Before the start of each set of exercise, Anand will place \(4\) blue balls and \(5\) green balls, identical apart from their colour, into a bag. Charlie will then randomly select \(4\) balls from the bag without replacement.

\(X\) denotes the number of blue balls chosen by Charlie in one set of exercise.

For each blue ball selected, Anand will perform \(5\) abdominal crunches.

For each green ball selected, Charlie will perform \(3\) chin-ups.

1. Show that \(\mathrm{P}\,(X=3)=\frac{10}{63}\) and tabulate the probability distribution of \(X\).[3]

\(C\) denotes the number of chin-ups performed by Charlie in \(1\) set of exercise.

2. Without using a graphing calculator, show that \(\mathrm{Var}(X)=\frac{50}{81}\) and, hence or otherwise, find \(\mathrm{Var}(C)\).[5]
3. Anand and Charlie decide to do \(3\) sets of exercise per day for a period of \(30\) days. Find the approximate probability that during this period, the total number of abdominal crunches performed by Anand is at least \(240\) greater than the total number of chin-ups performed by Charlie.[4]