2012 MJC P2 Q12
(a)
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 $\text{P}(X=k)$ by ${{p}_{k}}$, show that
$\frac{{{p}_{k+1}}}{{{p}_{k}}}=\frac{(n-k)p}{(k+1)(1-p)},\text{ }k=0,\text{ }1,\text{ }2,\text{ }…,\text{ }n-1.$
Hence find the most probable number of successes when $n=10$ and $p=\frac{1}{3}$.
[4]
(b)
In a certain country, it is known that $30%$ of the adult population has some knowledge of a foreign language.
(i) Find the probability that, in a random sample of $8$ adults, at most $2$ have some knowledge of a foreign language.
[1]
(ii) $400$ 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]
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