8 In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.
- State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means.
Assume now that a Poisson model is appropriate.
- Find the probability that in 10 ml there are at least 5 bacteria.
- Find the probability that in 3.7 ml there are exactly 2 bacteria.
- Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation.