2018 NYJC MYE P2 Q7

A bag contains \(6\) counters labelled ‘R’ and \(3\) counters labelled ‘G’. In a game, a player draws counters at random from the bag, one at a time with replacement, until \(4\) counters have been drawn. Let \(X\) be the number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. For example, the sequences ‘RGGR’, ‘GGRR’ or ‘GRRR’ will give \(X=1\) and the sequence ‘GRGR’ will give \(X=2\).

1. Show that \(\mathrm{P}(X=1)=\frac{46}{81}\) and tabulate the probability distribution of \(X\).[5]
2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\).[2]

The player loses \(\$2\) if he does not have a counter labelled ‘R’ drawn immediately after a counter labelled ‘G’ is drawn out of the \(4\) draws. Otherwise, the player wins \(\$1.50x\) for \(x\) number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. Alvin and Bernard each plays one game. Find the probability that their total winnings is at least $\\(4\).[3]