2022 ASRJC P2 Q10

A bag contains nine numbered discs. Three discs are numbered \(3\), four discs are numbered \(4\) and two discs are numbered \(-1\). Two discs are drawn simultaneously. The sum of numbers on them, denoted by \(X\), is recorded.

1. Find the probability distribution for \(X\).[3]
2. Find \(\mathrm{E}\left( X \right)\) and \(\mathrm{Var}\left( X \right)\).[2]
3. Two independent observations of \(X\) are taken. Find the probability that the difference between these two values is at most \(5\).[3]
4. Fifty independent observations of \(X\) are taken. Find the approximate probability that the sum of these fifty observations is between \(250\) and \(260\).[3]