Single period normal approximation - scaled period (normal approximation only)

Questions where the rate is scaled up to a longer period and the question asks only for the normal approximation probability, without requiring exact Poisson calculations for a sub-period.

27 questions · Moderate -0.2

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CAIE S2 2002 June Q6
10 marks Standard +0.3
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 2022 June Q5
9 marks Moderate -0.3
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 2024 June Q5
8 marks Moderate -0.8
5 Sales of cell phones at a certain shop occur singly, randomly and independently.
  1. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
    The average number of sales per hour is 1.2 .
    Assume now that a Poisson distribution is a suitable model.
  2. Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
  3. Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150 .
CAIE S2 2022 November Q3
7 marks Standard +0.3
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 2007 June Q5
8 marks Moderate -0.8
5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.
  1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
  2. Find the probability that exactly 8 people call in any half hour.
  3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.
CAIE S2 2010 June Q7
11 marks Standard +0.3
7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).
  1. State two assumptions required for the Poisson model to be valid.
  2. Find the probability that
    1. at least 1 patient will arrive in a 15-minute period,
    2. fewer than 4 patients will arrive in a 10-minute period.
    3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.
CAIE S2 2013 June Q6
7 marks Standard +0.3
6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be
  1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
  2. fewer than 1000 calls in four weeks ( 672 hours).
CAIE S2 2015 June Q6
9 marks Moderate -0.8
6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are
  1. exactly 2 errors in 10 pages,
  2. at least 3 errors in 6 pages,
  3. fewer than 50 errors in 200 pages.
CAIE S2 2018 June Q6
10 marks Standard +0.3
6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.
  1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
    Assume now that a Poisson distribution is a suitable model.
  2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
  3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
  4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).
CAIE S2 2017 March Q7
11 marks Moderate -0.3
7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 5.2 )\).
  1. State two assumptions required for the Poisson model to be valid in this context.
  2. (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,
    (b) Find the probability that more than 3 planes will arrive in a 40-minute period.
  3. The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.
CAIE S2 2019 November Q2
6 marks Standard +0.3
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
6 marks Standard +0.3
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 2011 November Q6
10 marks Moderate -0.8
6 Customers arrive at an enquiry desk at a constant average rate of 1 every 5 minutes.
  1. State one condition for the number of customers arriving in a given period to be modelled by a Poisson distribution. Assume now that a Poisson distribution is a suitable model.
  2. Find the probability that exactly 5 customers will arrive during a randomly chosen 30 -minute period.
  3. Find the probability that fewer than 3 customers will arrive during a randomly chosen 12-minute period.
  4. Find an estimate of the probability that fewer than 30 customers will arrive during a randomly chosen 2-hour period.
CAIE S2 2016 November Q3
7 marks Standard +0.3
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 S2 2012 June Q4
11 marks Moderate -0.3
4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.
  1. State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
  2. Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
  3. Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
  4. Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.
OCR MEI S2 2012 January Q2
19 marks Moderate -0.3
2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .
  1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
  2. A page is chosen at random. Find the probability of
    (A) exactly 1 error on this page,
    (B) at least 2 errors on this page. 10 pages are chosen at random.
  3. Find the probability of exactly 10 errors in these 10 pages.
  4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
  5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.
OCR Further Statistics AS 2021 November Q8
11 marks Standard +0.3
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.
    \section*{END OF QUESTION PAPER}
Edexcel S2 2016 June Q1
11 marks Standard +0.3
  1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.
The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75
  1. Find the largest possible value of \(v\)
  2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
  3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
  4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
Edexcel S2 2003 January Q3
12 marks Easy -1.2
3. A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.
  1. Write down two conditions that must apply for this model to be applicable. Assuming this model and a mean of 0.7 weeds per \(\mathrm { m } ^ { 2 }\), find
  2. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
  3. Using a suitable approximation, find the probability that in a plot of \(100 \mathrm {~m} ^ { 2 }\) there will be more than 66 of these weeds.
    (6)
Edexcel S2 2010 January Q5
9 marks Standard +0.3
  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
11 marks Moderate -0.3
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 2005 June Q3
14 marks Easy -1.2
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
  1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
  2. a randomly chosen page has no misprints,
  3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
  4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
Edexcel S2 2013 June Q2
10 marks Moderate -0.3
  1. The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25
Find the probability that
  1. a randomly chosen metre of cloth has 1 defect,
  2. the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 A tailor buys 300 metres of cloth.
  3. Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects.
Edexcel S2 Q6
18 marks Standard +0.8
6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that
  1. at least 4 customers arrive in the next 10 minutes,
  2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
  3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
  4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END
OCR MEI Further Statistics A AS 2022 June Q2
7 marks Easy -1.2
2 On a car assembly line, a robot is used for a particular task.
  1. State the conditions under which a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week. It is given that the average number of breakdowns of the robot in a week is 1.7 . For the remainder of this question, you should assume that a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week.
    1. Find the probability that the number of breakdowns of the robot in a week is exactly 4.
    2. Determine the probability that the number of breakdowns of the robot in a week is at least 2 .
  3. Determine the probability that the number of breakdowns of the robot in 52 weeks is less than 100.