2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water.
Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.
- Use a Poisson distribution to
(A) find the probability that a 5 ml sample contains exactly 2 bacteria,
(B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 . - The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria.
The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.
- If the 5 ml sample contains more than 2 bacteria, then a 50 ml sample is taken.
- If this 50 ml sample contains more than 8 bacteria, then a sample of 1000 ml is taken.
- If this 1000 ml sample contains more than 90 bacteria, then the supply is declared to be 'questionable'.
- Find the probability that a random sample of 50 ml contains more than 8 bacteria.
- Use a suitable approximating distribution to find the probability that a random sample of 1000 ml contains more than 90 bacteria.
- Find the probability that the supply is declared to be questionable.