2 It was stated in 2012 that \(3 \%\) of \(\pounds 1\) coins were fakes. Throughout this question, you should assume that this is still the case.
- Find the probability that, in a random selection of \(25 \pounds 1\) coins, there is exactly one fake coin.
A random sample of \(250 \pounds 1\) coins is selected.
- Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
- Use a Poisson distribution to find the probability that, in this sample, there are
(A) exactly 10 fake coins,
(B) at least 10 fake coins. - Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins.
It is known that \(0.2 \%\) of another type of coin are fakes.
- A random sample of size \(n\) of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }\), where \(\lambda = 0.002 n\).
- Use the approximation \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }\) for small values of \(\lambda\) to estimate the value of \(n\) for which the probability in part ( \(\mathbf { v }\) ) is equal to 0.995 .