N2020 P2 Q8

N2020 P2 Q8

Junior College 2
7 marks
Free

In a game, a computer randomly chooses \(12\) shapes from \(11\) circles and \(17\) rectangles. The number of rectangles chosen is denoted by \(R\).

  1. Show that \(P(R=1)<P(R=2)\).[2]
  2. The number of rectangles available is now increased by \(r\). The computer randomly chooses \(12\) shapes from the \(11\) circles and \((17+r)\) rectangles. The probability that \(4\) rectangles are chosen is now \(15\) times the probability that \(3\) rectangles are chosen. Find the value of \(r\).[5]

Default solution

(i) \(P(R=1)=\frac{\binom{17}{1}\binom{11}{11}}{\binom{28}{12}}\) and \(P(R=2)=\frac{\binom{17}{2}\binom{11}{10}}{\binom{28}{12}}\). Their ratio is \(P(R=2)/P(R=1)=88>1\).

(ii) \(\frac{P(R=4)}{P(R=3)}=\frac{\binom{17+r}{4}\binom{11}{8}}{\binom{17+r}{3}\binom{11}{9}}=\frac{3(14+r)}4=15\). Hence \(r=6\).

Answer:(ii) \(r=6\)

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