2023 ACJC P2 Q9

A bag contains one green disc, \(r\) red discs and \(2r\) blue discs, where \(r>1\).

1. Tim and Sim play a game by taking turns to draw a disc from the bag at random. The drawn disc is replaced after each turn. The first person to draw the green disc wins. If Sim started the game first, show that the probability of Sim winning the game is \(\frac{3r+1}{6r+1}\).[2]
2. In another game, a player draws \(2\) discs from the bag at random. If he draws a red disc, he scores \(5\) points, if he draws a blue disc, he scores \(2\) points and if he draws a green disc he scores \(0\). The player’s score, denoted by \(T\), is found by taking the difference in the number of points he scores for the two discs.
   1. Find \(\mathrm{P}\left( T=t \right)\) for all possible values of \(t\), leaving your answers in terms of \(r\).[3]
   2. Show that the expectation of the player’s score is \(\frac{2\left( 2r+3 \right)}{3r+1}\).[1]
3. In a third game, a player draws a disc from the bag at random. If he draws a green disc, he wins \(\$0.15\), if he draws a blue disc, he wins \({\$0.10}\) and if he draws a red disc, he loses \(\$0.25\). May and Jun each draw a disc from the bag once, with the disc replaced in the bag after the first player drew. The probability that May ends up with more money than Jun is \(\frac{27}{112}\).
   Find the value of \(r\).[2]
   Is the game fair for a player? Explain.[1]