$3$ discs are taken, at random and without replacement, from a bag containing $5$ red discs and

Question

$3$ discs are taken, at random and without replacement, from a bag containing $5$ red discs and $n$ white discs, where $n \ge 3$. The random variable $R$ is the number of red discs taken and the random variable $W$ is the number of white discs taken. Determine the probability distribution of $W$.[3] Show that $\mathrm{E}(W) = \frac{3n}{n+5}$ and $\mathrm{Var}(W) = \frac{g(n)}{(n+5)^2 (n+4)}$, where $g(n)$ is a quadratic polynomial to be determined.[5] Hence write down an expression for $\mathrm{Var}(R)$.[1]

Accepted Answer

(i) Distribution shown (ii) $g(n)=15n^2+30n$ (iii) $\mathrm{Var}(R)=\dfrac{15n(n+2)}{(n+5)^2(n+4)}$

2021 ACJC P2 Q8

2021 ACJC P2 Q8

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