6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.
- For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
- For another path the mean of \(X\) is 1.3.
(a) On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
(b) Find the probability that there are more than 1250 worms on a one kilometre length of this path.