Scaled time period probability

Questions requiring scaling of the Poisson mean to a different time period (e.g., from per week to per 2 weeks, or per minute to per 90 minutes) before calculating probabilities.

9 questions

CAIE S2 2013 June Q6
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).
Edexcel S2 2021 June Q2
  1. Luis makes and sells rugs. He knows that faults occur randomly in his rugs at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)
    1. Find the probability of there being exactly 5 faults in one of his rugs that is \(4 \mathrm {~m} ^ { 2 }\) in size.
    2. Find the probability that there are more than 5 faults in one of his rugs that is \(6 \mathrm {~m} ^ { 2 }\) in size.
    Luis makes a rug that is \(4 \mathrm {~m} ^ { 2 }\) in size and finds it has exactly 5 faults in it.
  2. Write down the probability that the next rug that Luis makes, which is \(4 \mathrm {~m} ^ { 2 }\) in size, will have exactly 5 faults. Give a reason for your answer. A small rug has dimensions 80 cm by 150 cm . Faults still occur randomly at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)
    Luis makes a profit of \(\pounds 80\) on each small rug he sells that contains no faults but a profit of \(\pounds 60\) on any small rug he sells that contains faults. Luis sells \(n\) small rugs and expects to make a profit of at least \(\pounds 4000\)
  3. Calculate the minimum value of \(n\) Luis wishes to increase the productivity of his business and employs Rhiannon. Faults also occur randomly in Rhiannon's rugs and independently to faults made by Luis. Luis randomly selects 10 small rugs made by Rhiannon and finds 13 faults.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the rate at which faults occur is higher for Rhiannon than for Luis. State your hypotheses clearly.
Edexcel S2 2014 June Q1
  1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
    1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
      1. exactly 7
      2. at least 10
    A patient arrives at 11.30 a.m.
  2. Find the probability that the next patient arrives before 11.45 a.m.
Edexcel S2 2015 June Q1
  1. In a survey it is found that barn owls occur randomly at a rate of 9 per \(1000 \mathrm {~km} ^ { 2 }\).
    1. Find the probability that in a randomly selected area of \(1000 \mathrm {~km} ^ { 2 }\) there are at least 10 barn owls.
    2. Find the probability that in a randomly selected area of \(200 \mathrm {~km} ^ { 2 }\) there are exactly 2 barn owls.
    3. Using a suitable approximation, find the probability that in a randomly selected area of \(50000 \mathrm {~km} ^ { 2 }\) there are at least 470 barn owls.
    4. The proportion of houses in Radville which are unable to receive digital radio is \(25 \%\). In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.
    5. Find \(\mathrm { P } ( 5 \leqslant X < 11 )\)
    A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.
  2. Test, at the \(10 \%\) level of significance, the radio company's claim. State your hypotheses clearly.
AQA S2 2013 June Q4
4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
  1. Find the probability that the satellite detects:
    1. exactly 4 GRBs during one particular week;
    2. at least 2 GRBs on one particular day;
    3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
  2. Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
    (1 mark)
SPS SPS FM Statistics 2023 January Q1
1. Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
  1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. Indre is allowed 20 minutes of break time during each 4 -hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
  2. Find the probability that in exactly 3 of these periods there were no calls.
OCR S2 Q1
1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.
  1. Use a Poisson distribution to find the probability that, at a given moment,
    (a) in a randomly chosen area of 3 acres there are at least 4 foxes,
    (b) in a randomly chosen area of 1 acre there are exactly 2 foxes.
  2. Explain briefly why a Poisson distribution might not be a suitable model.
OCR S2 2010 January Q9
9 Buttercups in a meadow are distributed independently of one another and at a constant average incidence of 3 buttercups per square metre.
  1. Find the probability that in 1 square metre there are more than 7 buttercups.
  2. Find the probability that in 4 square metres there are either 13 or 14 buttercups.
  3. Use a suitable approximation to find the probability that there are no more than 69 buttercups in 20 square metres.
  4. (a) Without using an approximation, find an expression for the probability that in \(m\) square metres there are at least 2 buttercups.
    (b) It is given that the probability that there are at least 2 buttercups in \(m\) square metres is 0.9 . Using your answer to part (a), show numerically that \(m\) lies between 1.29 and 1.3.
AQA Further AS Paper 2 Statistics 2023 June Q6
6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6
  1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
  2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
    6
  3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6
    1. Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
      6
  5. (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.