Discrete Random Variables (DRV)

Tom has a bag of wooden rectangular blocks of identical size. The bag contains \(1\) blue block, \(m\) red blocks and \((m-1)\) yellow blocks, where \(m > 2\). Tom and Jerry play a game using Tom's bag of wooden blocks. Jerry draws \(2\) blocks at random, one at a time, without replacement. \(3\) points will be awarded if a yellow block is drawn, \(2\) points will be awarded if a red block is drawn, but no points will be awarded if a blue block is drawn. Jerry's final score is the product of the points awarded for the \(2\) blocks drawn.

Let the random variable \(X\) denotes Jerry's final score.

1. Show that \(P(X=0) = \frac{1}{m}\) and hence, find the probability distribution of \(X\).[4]
2. Find the value of \(m\) if Jerry's expected final score is \(5\).[2]

Tom pays Jerry \$\(5\) if Jerry's final score is at least \(5\) and Jerry pays Tom \$\(a\) if his final score is less than \(5\).

3. Using the value of \(m\) found in (ii), find the range of values of \(a\) if Tom is expected to make a profit.[2]