2018 RI CT3 J2 Q13

A bag contains \(20\) blue balls and \(15\) red balls. In a game, Joe draws \(12\) balls one by one, at random and without replacement. The number of blue balls Joe draws from the bag is denoted by \(X\).

1. Find \(\mathrm{E}\left( X \right)\) and \(\mathrm{Var}\left( X \right)\).[4]
2. If \(\mathrm{P}\left( X=x \right)<\mathrm{P}\left( X=x+1 \right)\), show that \(x\) satisfies the inequality
   \(\left( x+a \right)\left( x+b \right)<\left( c-x \right)\left( d-x \right)\)
   where \(a\), \(b\), \(c\) and \(d\) are constants to be determined. Use this inequality to find the number of blue balls that Joe is most likely to draw.[5]

Joe plays this game \(30\) times.

3. Find the probability that Joe draws at least \(10\) blue balls in more than \(4\) of his \(30\) games.[3]