4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
\begin{enumerate}[label=(\roman*)]
\item Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,\\
(A) both seeds germinate,\\
(B) just one seed germinates,\\
(C) neither seed germinates.
\item Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
\item A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
\item The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only $85 \%$ successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds.

If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is $90 \%$. The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
\item Write down suitable hypotheses and carry out a test at the $5 \%$ level to determine whether there is any evidence to support the gardener's suspicions.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S1  Q4 [18]}}