The random variable $X$ is the number of successes in $n$ independent trials of an experiment in

Question

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at a single trial is $p$. Denoting $\mathrm{P}(X=k)$ by ${{p}_{k}}$, show that $\frac{{{p}_{k+1}}}{{{p}_{k}}}=\frac{(n-k)p}{(k+1)(1-p)},\mathrm{ }k=0,\mathrm{ }1,\mathrm{ }2,\mathrm{ }...,\mathrm{ }n-1.$ Hence find the most probable number of successes when $n=10$ and $p=\frac{1}{3}$.[2] In a certain country, it is known that $30\%$ of the adult population has some knowledge of a foreign language. Find the probability that, in a random sample of $8$ adults, at most $2$ have some knowledge of a foreign language.[1] $400$ adults are chosen at random. Use a suitable approximation to find the least value of $n$ so that the probability that less than $n$ adults having some knowledge of a foreign language is at least $0.9$.[4] For one particular foreign language, $99\%$ of the adult population does not have some knowledge of it. Using a suitable approximation, find the probability that, in a random sample of $400$ adults, more than $395$ do not have some knowledge of the particular foreign language.[3]

Accepted Answer

(a) $\frac{p_{k+1}}{p_k}=\frac{(n-k)p}{(k+1)(1-p)}$, so the mode is $3$ (b)(i) $0.552$ (b)(ii) $n=133$ (b)(iii) $0.629$

2012 MJC P2 Q12

2012 MJC P2 Q12

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