N2016 P2 Q10

Mia owns a field. Various types of weed are found in Mia's field.

1. State, in this context, two conditions that must be met for the numbers of a particular type of weed in Mia's field to be well modelled by a Poisson distribution.[2]

For the remainder of this question assume that these conditions are met.

There is an average of \(1.5\) dandelion plants (a type of weed) per \(\text{m}^2\) in Mia's field.

2. Find the probability that in \(1\text{ m}^2\) of Mia's field there are at least \(2\) dandelion plants.[2]
3. Find the probability that in \(4\text{ m}^2\) of Mia's field there are at most \(3\) dandelion plants.[2]
4. Use a suitable approximation, which should be stated, to find the probability that the number of dandelion plants in an \(80\text{ m}^2\) area of Mia's field is between \(110\) and \(140\) inclusive.[4]

The distribution of daisies (another type of weed) per \(\text{m}^2\) in Mia's field can be modelled by \(\mathrm{Po}(\lambda)\). The probability that the number of daisies in a \(1\text{ m}^2\) area of the field is less than or equal to \(2\) is the same as the probability that the number of daisies in a \(2\text{ m}^2\) area of the field is more than \(2\).

5. Write down an equation in \(\lambda\) and solve it to find \(\lambda\).[4]