Single period normal approximation

Questions that use normal approximation to find probability for a Poisson distribution over a single time period (e.g., one year, one day, one hour), without comparing multiple periods or using binomial structure.

18 questions

CAIE S2 2002 June Q6
6 Between 7 p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour.
  1. Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive.
  2. A patient arrives at exactly 10.15 p.m. Find the probability that at least one more patient arrives before 10.35 p.m.
  3. Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11 p.m. on any particular night.
CAIE S2 2021 June Q7
7 Customers arrive at a particular shop at random times. It has been found that the mean number of customers who arrive during a 5 -minute interval is 2.1 .
  1. Find the probability that exactly 4 customers arrive during a 10 -minute interval.
  2. Find the probability that at least 4 customers arrive during a 20 -minute interval.
  3. Use a suitable approximating distribution to find the probability that fewer than 40 customers arrive during a 2-hour interval.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2022 June Q5
5 The number of clients who arrive at an information desk has a Poisson distribution with mean 2.2 per 5-minute period.
  1. Find the probability that, in a randomly chosen 15 -minute period, exactly 6 clients arrive at the desk.
  2. If more than 4 clients arrive during a 5 -minute period, they cannot all be served. Find the probability that, during a randomly chosen 5 -minute period, not all the clients who arrive at the desk can be served.
  3. Use a suitable approximating distribution to find the probability that, during a randomly chosen 1-hour period, fewer than 20 clients arrive at the desk.
CAIE S2 2022 November Q3
3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.
  1. Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
  2. Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.
CAIE S2 2023 November Q3
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.
    2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.
      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 ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
    1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
    2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.
CAIE S2 2016 March Q6
6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.
  1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
  2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
    (b) Justify the approximating distribution used in part (ii)(a).
  3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .
CAIE S2 2019 November Q2
2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.
  1. Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.
  2. Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.
CAIE S2 2009 November Q2
2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.
  1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
  2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.
CAIE S2 2014 November Q6
6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of
  1. exactly 4 calls in an 8 -minute period,
  2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
  3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.
CAIE S2 2016 November Q3
3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that
  1. more than 3 particles are emitted during a 20 -second period,
  2. more than 240 particles are emitted during a 1-hour period.
OCR MEI S2 2013 January Q2
2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.
  1. Find the probability that John observes
    (A) no butterflies in a minute,
    (B) at least 2 butterflies in a minute,
    (C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
  2. Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
  3. Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
  4. Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.
OCR Further Statistics AS 2021 November Q8
8
  1. A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
  2. The random variable \(X\) represents the number of particles emitted by a substance in a fixed time interval \(t\) minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for \(X\) if the value of \(t\) is very large.
  3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that
    \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = \mathrm { P } ( \mathrm { Y } = \mathrm { r } + 1 )\)
    \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = 1.5 \times \mathrm { P } ( \mathrm { Y } = \mathrm { r } - 1 )\). Determine the following, in either order.
    • The value of \(r\)
    • The value of \(\lambda\)
    \section*{END OF QUESTION PAPER}
Edexcel S2 2010 January Q5
  1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
Find the probability that
  1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
  2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
Edexcel S2 2013 January Q2
2. In a village, power cuts occur randomly at a rate of 3 per year.
  1. Find the probability that in any given year there will be
    1. exactly 7 power cuts,
    2. at least 4 power cuts.
  2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20
Edexcel S2 Q5
5. 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.
  2. Find the probability that a particular page has more than 2 misprints.
  3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. Chapter 2 is longer, at 20 pages.
  4. 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. \section*{STATISTICS 2 (A) TEST PAPER 5 Page 2}
Edexcel S2 Q4
4. A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that
  1. he is not late at all,
  2. he is late more than twice. In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.
  3. Use the Normal approximation to estimate the probability that he loses his privileges. \section*{STATISTICS 2 (A)TEST PAPER 6 Page 2}
Edexcel S2 Q4
4. It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3 . Find the probability that
  1. there will be no failures in a one-hour period,
  2. there will be more than 4 failures in a 30 -minute period. Using a suitable approximation, find the probability that in a 24-hour period there will be
  3. less than 60 failures,
  4. exactly 72 failures.
Edexcel S2 Q5
5. A charity receives donations of more than \(\pounds 10000\) at an average rate of 25 per year. Find the probability that the charity receives
  1. exactly 30 such donations in one year,
  2. less than 3 such donations in one month.
  3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than \(\pounds 10000\) in the next two years.