Joint probability of separate processes

Questions asking for the probability that specific events occur in two or more independent Poisson processes separately (e.g., at least 2 men AND at least 3 women), requiring multiplication of individual Poisson probabilities.

4 questions · Standard +0.0

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OCR Further Statistics AS 2023 June Q1
7 marks Moderate -0.8
1 A radar device is used to detect flaws in motorway roads before they become dangerous. The number of flaws in a 1 km stretch of motorway is denoted by \(X\). It may be assumed that flaws occur randomly.
  1. State two further assumptions that are necessary for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be modelled by distribution \(\operatorname { Po } ( 5.7 )\).
  2. Determine the probability that in a randomly chosen stretch of motorway, of length 1 km , there are between 8 and 11 flaws, inclusive.
  3. Determine the probability that in two randomly chosen, non-overlapping, stretches of motorway, each of length 5 km , there are at least 30 flaws in one stretch and fewer than 30 flaws in the other stretch.
Edexcel S2 2012 January Q5
7 marks Moderate -0.3
  1. The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
    1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box.
    A retailer buys 2 boxes of components.
  2. Estimate the probability that there are at least 4 defective components in each box.
CAIE S2 2011 November Q7
11 marks Standard +0.8
The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.
  1. Find the probability that, in a half-hour period,
    1. 2 or more men and 1 or more women will visit the clinic, [4]
    2. a total of 3 or more people will visit the clinic. [3]
  2. Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]
AQA S2 2016 June Q1
13 marks Standard +0.3
The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:
  1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
  2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
  3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
  4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
  5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]