Combined Poisson approximation and exact calculation

A question is this type if and only if it requires both exact Poisson probability calculations (using tables or formula) and normal approximation in different parts of the same question.

4 questions

CAIE S2 2024 June Q1
1 A random variable \(X\) has the distribution \(\mathrm { Po } ( 145 )\).
  1. Use a suitable approximating distribution to calculate \(\mathrm { P } ( X \leqslant 150 )\).
  2. Justify the use of your approximating distribution in this case.
OCR S2 2008 January Q6
6 The number of house sales per week handled by an estate agent is modelled by the distribution \(\operatorname { Po } ( 3 )\).
  1. Find the probability that, in one randomly chosen week, the number of sales handled is
    (a) greater than 4 ,
    (b) exactly 4 .
  2. Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.
  3. One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.
    (a) Explain, in non-technical language, what you understand by this condition.
    (b) State another condition that is needed.
OCR S2 2011 June Q7
7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).
  1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
  2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
  3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.
AQA S3 2013 June Q6
6 The demand for a WWSatNav at a superstore may be modelled by a Poisson distribution with a mean of 2.5 per day. The superstore is open 6 days each week, from Monday morning to Saturday evening.
    1. Determine the probability that the demand for WWSatNavs during a particular week is at most 18 .
    2. The superstore receives a delivery of WWSatNavs on each Sunday evening. The manager, Meena, requires that the probability of WWSatNavs being out of stock during a week should be at most \(5 \%\). Determine the minimum number of WWSatNavs that Meena requires to be in stock after a delivery.
    1. Use a distributional approximation to estimate the probability that the demand for WWSatNavs during a period of \(\mathbf { 2 }\) weeks is more than 35.
    2. Changes to the superstore's delivery schedule result in it receiving a delivery of WWSatNavs on alternate Sunday evenings. Meena now requires that the probability of WWSatNavs being out of stock during the 2 weeks following a delivery should be at most \(5 \%\). Use a distributional approximation to determine the minimum number of WWSatNavs that Meena now requires to be in stock after a delivery.
      (3 marks)