Justify Poisson approximation only

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

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.

1. 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.
2. Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
3. 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.
4. 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.
5. 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 \(\lambda = 0.002 n\).
6. 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 .

2 On average 1 in 4000 people have a particular antigen in their blood (an antigen is a molecule which may cause an adverse reaction).

1. 1. A random sample of 1200 people is selected. The random variable \(X\) represents the number of people in the sample who have this antigen in their blood. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
   2. Use either a binomial or a Poisson distribution to calculate each of the following probabilities.
      • \(\mathrm { P } ( X = 3 )\)
2. \(\mathrm { P } ( X > 3 )\)
3. A researcher needs to find 2 people with the antigen. Find the probability that at most 5000 people have to be tested in order to achieve this.

1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows

$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$

1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)