2022 MI P2 Q7
2022 MI P2 Q7
8 marks
Prelims
A bag contains three red balls, \(n-1\) blue balls and \(n\) white ball, where \(n\ge 3\). The balls are identical except for their colour. Two balls are drawn at random, without replacement. Two points is given for each red ball drawn, one point is given for each blue ball drawn and one point is deducted for each white ball drawn. Let \(X\) be the score obtained from adding the points of the two balls.
- Show that \(\mathrm{P}\left( X=0 \right)=\frac{n\left( n-1 \right)}{\left( n+1 \right)\left( 2n+1 \right)}\).[2]
- Find the probability distribution of \(X\).[3]
- Given that the average score of picking the two balls is at least \(0.16\), find the largest possible number of white balls in the bag.[3]
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Answer:(ii) \({\mathrm{P}(X = -2)\frac{n(n-1)}{2(n+1)(2n+1)}}\); \({\mathrm{P}(X = 0) = \frac{n(n-1)}{(n+1)(2n+1)}}\); \({\mathrm{P}(X = 1) = \frac{3n}{(n+1)(2n+1)}}\); \({\mathrm{P}(X = 2) = \frac{(n-1)(n-2)}{2(n+1)(2n+1)}}\); \({P(X = 3) = \frac{3(n-1)}{(n+1)(2n+1)}}\); \({\mathrm{P}(X = 4) = \frac{3}{(n+1)(2n+1)}}\) (iii) Largest possible number of white balls is \({30}\).
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