Multiple periods with binomial structure

Questions that first use normal approximation for a Poisson distribution, then apply this to multiple independent periods using binomial probability (e.g., probability that x out of n weeks satisfy a condition).

3 questions · Standard +0.3

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities
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Edexcel S2 2022 January Q1
11 marks Standard +0.3
1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528
  1. Find the value of \(x\)
  2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
  3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".
Edexcel S2 2012 June Q4
14 marks Standard +0.3
4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.
  1. Find the probability that in the next 4 weeks the estate agent sells,
    1. exactly 3 houses,
    2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
  2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
  3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.
Edexcel S2 2018 June Q1
12 marks Standard +0.3
  1. In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9
    1. Find
      1. \(\mathrm { P } ( X > 5 )\)
      2. \(\mathrm { P } ( 4 \leqslant X < 10 )\)
    The length of a working day is 7 hours.
  2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
  3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.