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.

6 questions · Standard +0.3

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CAIE S2 2024 June Q1
5 marks Standard +0.3
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
11 marks Standard +0.3
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
    1. greater than 4 ,
    2. exactly 4 .
    3. 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.
    4. 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
14 marks Standard +0.8
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
11 marks Standard +0.3
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)
Pre-U Pre-U 9795/2 2011 June Q2
8 marks Moderate -0.3
2 The discrete random variable \(X\) has a Poisson distribution with mean 12.25.
  1. Calculate \(\mathrm { P } ( X \leqslant 5 )\).
  2. Calculate an approximate value for \(\mathrm { P } ( X \leqslant 5 )\) using a normal approximation to the Poisson distribution.
  3. Comment, giving a reason, on the accuracy of using a normal approximation to the Poisson distribution in this case.
Edexcel S2 2011 January Q6
16 marks Standard +0.3
Cars arrive at a motorway toll booth at an average rate of 150 per hour.
  1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute. [2]
  2. State clearly any assumptions you have made by suggesting this model. [2]
Using your model,
  1. find the probability that in any given minute
    1. no cars arrive,
    2. more than 3 cars arrive.
    [3]
  2. In any given 4 minute period, find \(m\) such that P(\(X > m\)) = 0.0487 [3]
  3. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period. [6]