Binomial of Poisson approximations

A question is this type if and only if it involves using a binomial distribution where each trial's outcome is itself modeled by a Poisson approximation (e.g., number of days with certain events).

3 questions

CAIE S2 2012 November Q7
7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).
  1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
  2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
  3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.
OCR S2 2007 January Q5
5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)
  1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
  2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
  3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.
OCR S2 Specimen Q6
6 On average a motorway police force records one car that has run out of petrol every two days.
  1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
    (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
  2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.