Single period normal approximation - scaled period (exact Poisson part)

Questions where the rate must be scaled to a longer period, and the question includes finding exact Poisson probabilities for a shorter sub-period before applying normal approximation to the scaled period.

2 questions · Standard +0.3

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CAIE S2 2023 November Q3
10 marks Standard +0.3
A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.
  1. Assume that the owner is correct.
    1. Find the probability that there will be at least 4 hits during a 10-minute period. [3]
    2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]
A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).
    1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution. [1]
    2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]
Edexcel S2 Q5
13 marks Standard +0.3
A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,
  1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page. [1 mark]
  2. Find the probability that a particular page has more than 2 misprints. [3 marks]
  3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]
Chapter 2 is longer, at 20 pages.
  1. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. [7 marks]