Calculate multiple probabilities using Poisson approximation

Questions that require calculating two or more different probability values in separate parts (e.g., both P(X<a) and P(X=b)) using Poisson approximation to the binomial.

6 questions · Moderate -0.6

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CAIE S2 2021 March Q4
10 marks Moderate -0.3
4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).
  1. State the distribution of \(X\), giving the values of any parameters.
  2. State an approximating distribution for \(X\), giving the values of any parameters.
  3. Use this approximating distribution to find each of the following.
    1. \(\mathrm { P } ( X = 4 )\).
    2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
  4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.
CAIE S2 2022 November Q3
6 marks Moderate -0.8
1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.
  1. Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
  2. Justify your approximating distribution. [2]
CAIE S2 2016 June Q3
6 marks Moderate -0.3
1\% of adults in a certain country own a yellow car.
  1. Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car. [4]
  2. Justify your approximation. [2]
Edexcel S2 2004 January Q4
10 marks Moderate -0.8
  1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]
A researcher has suggested that 1 in 150 people is likely to catch a particular virus. Assuming that a person catching the virus is independent of any other person catching it,
  1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
  2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]
Edexcel S2 2009 January Q5
9 marks Moderate -0.3
A factory produces components of which 1\% are defective. The components are packed in boxes of 10. A box is selected at random.
  1. Find the probability that the box contains exactly one defective component. [2]
  2. Find the probability that there are at least 2 defective components in the box. [3]
  3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]
Edexcel S2 Q6
13 marks Moderate -0.8
A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
  1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
  2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
  3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
  4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]