2024 NYJC P2 Q7

A computer randomly chooses \(10\) shapes from \(18\) squares and \(12\) triangles. The number of squares chosen is denoted by \(S\).

1. The most probable number of squares that the computer chooses is denoted by \(s\). By using the fact that \(P\left( S=s \right)>P\left( S=s+1 \right)\), show that \(s\) satisfies the inequality
   \(\left( s+1 \right)\left( s+3 \right)>\left( a-s \right)\left( b-s \right)\)
   where \(a\) and \(b\) are constants to be determined. Hence find the value of \(s\).[4]

In a game, the computer randomly chooses \(6\) squares and \(4\) triangles. The squares are numbered from \(1\) to \(6\) and the triangles are numbered from \(1\) to \(4\). A square and a triangle are randomly chosen. Let \(X\) be the score which is defined as the difference (taken as always positive) between the numbers of the chosen square and triangle.

2. Find the probability distribution of \(X\).[3]

Three independent games are played. The total score of the games is \(3\).

3. Find the probability that the score of the first game is \(2\).[3]