2022 NYJC P2 Q9
2022 NYJC P2 Q9
11 marks
Prelims
A bag contains three yellow marbles, one blue marble and \(x\) red marbles, where \(x>1\). In a game, Lily takes \(~2\) marbles at random from the bag, without replacement. She scores \(~2\) points for each red marble taken and \(1\) point for each yellow marble taken. If a blue marble is taken, she loses \(~2\) points. The random variable \(S\) is the sum of the points of the two marbles taken.
- Find \(\mathrm{P}\left( S=s \right)\) for all possible values of \(s\).[4]
- Show that \(\mathrm{E}\left( S \right)=\frac{2\left( 2x+1 \right)}{x+4}\) and find an expression for \(\mathrm{Var}\left( S \right)\).[5]
- Find the least value of \(x\) if \(\mathrm{E}\left( S \right)>2\).[2]
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Answer:(i) \({\mathrm{P}(S=-1)=\frac{6}{(x+4)(x+3)}}\); \({\mathrm{P}(S=0)=\frac{2x}{(x+4)(x+3)}}\); \({\mathrm{P}(S=2)=\frac{6}{(x+4)(x+3)}}\); \({\mathrm{P}(S=3)=\frac{6x}{(x+4)(x+3)}}\); \({\mathrm{P}(S=4)=\frac{x(x-1)}{(x+4)(x+3)}}\) (ii) \({\mathrm{Var}(S)=\frac{2(19x+27)(x+2)}{(x+4)^2 (x+3)}}\) (iii) Least \({x=4}\)
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