Poisson approximation justification or comparison

5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).

1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
   Use your approximating distribution to answer parts (b) and (c).
2. Find \(\mathrm { P } ( X \leqslant 3 )\).
3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
   The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).

2 Almost all plants of a particular species have red flowers. However on average 1 in every 1500 plants of this species have white flowers. A random sample of 2000 plants of this species is selected. The random variable \(X\) represents the number of plants in the sample that have white flowers.

1. Name two distributions which could be used to model the distribution of \(X\), stating the parameters of each of these distributions. You may use either of the distributions you have named in the rest of this question.
2. Calculate each of the following.
   • \(\mathrm { P } ( X = 2 )\)
   • \(\mathrm { P } ( X > 2 )\)
   • A random sample of 20000 plants of this species is selected.
   Calculate the probability that there are at least 10 plants in the sample that have white flowers.

4 At a parcel delivery company it is known that the probability that a parcel is delivered to the wrong address is 0.0005 . On a particular day, 15000 parcels are delivered. The number of parcels delivered to the wrong address is denoted by the random variable \(X\).

1. Explain why the binomial distribution and the Poisson distribution could both be suitable models for the distribution of \(X\).
2. Use a Poisson distribution to find each of the following.
   • \(\mathrm { P } ( X = 5 )\)
   • \(\mathrm { P } ( X \geqslant 8 )\)
   You are given that 15000 parcels are delivered each day in a 5-day working week.
3. 1. Determine the probability that at least 40 parcels are delivered to the wrong address during the week.
   2. Determine the probability that at least 8 parcels are delivered to the wrong address on each of the 5 days in the week.

1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).

1. Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
2. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
3. Use a Poisson distribution to calculate each of the following probabilities.
   • \(\mathrm { P } ( X = 10 )\)
   • \(\mathrm { P } ( X > 10 )\)
   • In another challenge on the website, 50 people are each given 20 independent attempts to try to get four sixes as often as they can.
   Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.

4 A radioactive source contains 1000000 nuclei of a particular radioisotope. On average 1 in 200000 of these nuclei will decay in a period of 1 second. The random variable \(X\) represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\). Use a Poisson distribution to answer parts (b) and (c).
2. Calculate each of the following probabilities.
   • \(\mathrm { P } ( X = 6 )\)
   • \(\mathrm { P } ( X > 6 )\)
   • Determine an estimate of the probability that at least 60 nuclei decay in a period of 10 seconds.