Multiple independent time periods

Questions that involve calculating probabilities across multiple independent time periods (e.g., probability that events occur in each of several separate periods, or only one of several periods).

10 questions

CAIE S2 2009 June Q3
3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.
  1. Find the probability that more than 3 major avalanches occur in a 3-month period.
  2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
  3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.
OCR Further Statistics AS 2019 June Q2
2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
  3. State a necessary assumption for the validity of your calculation in part (b).
Edexcel S2 2015 June Q2
2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8
  1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
  2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
  3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.
Edexcel S2 2017 June Q1
  1. At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
    1. Find the probability that there are no signal failures on a randomly selected day.
    2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
    3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
    Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
  2. State the hypotheses for this test.
  3. Find the largest value of \(f\) for which the null hypothesis should be rejected.
Edexcel S2 2024 June Q1
1 A garage sells tyres. The number of customers arriving at the garage to buy tyres in a 10-minute period is modelled by a Poisson distribution with mean 2
  1. Find the probability that
    1. fewer than 4 customers arrive to buy tyres in the next 10 minutes,
    2. more than 5 customers arrive to buy tyres in the next 10 minutes. The manager randomly selects 20 non-overlapping, 30-minute periods.
  2. Find the probability that there are between 4 and 7 (inclusive) customers arriving to buy tyres in exactly 15 of these 30-minute periods. The manager believes that placing an advert in the local paper will lead to a significant increase in the number of customers arriving at the garage.
    A week after the advert is placed, the manager randomly selects a 25 -minute period and finds that 10 customers arrive at the garage to buy tyres.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the manager's belief.
    State your hypotheses clearly.
  4. Explain why the Poisson distribution is unlikely to be valid for the number of tyres sold during a 10-minute period.
Edexcel S2 2001 June Q2
2. On a stretch of motorway accidents occur at a rate of 0.9 per month.
  1. Show that the probability of no accidents in the next month is 0.407 , to 3 significant figures. Find the probability of
  2. exactly 2 accidents in the next 6 month period,
  3. no accidents in exactly 2 of the next 4 months.
Edexcel FS1 AS 2019 June Q3
  1. Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.
    1. Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work.
    Andreia asks the secretary to produce a 250 -word article for a magazine.
  2. Find the probability that there are exactly 5 errors in this article. Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words. In scheme \(\mathbf { A }\) the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors. In scheme \(\mathbf { B }\) the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56
  3. Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose. Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test.
Edexcel FS1 2019 June Q2
  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. On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
  3. Find the probability that Indre missed exactly 1 call in each of these 2 breaks.
AQA Further Paper 3 Statistics 2019 June Q8
8 The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes. 8
  1. Find the probability that:
    8
    1. the office receives exactly 2 calls in 10 minutes; 8
  3. (ii) the office receives more than 30 calls in an hour.
    8
  4. The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods. Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.
    8
  5. The office has just received a call.
    8
    1. Find the probability that the next call is received more than 10 minutes later.
      8
  7. (ii) Mahah arrives at the office 5 minutes after the last call was received.
    State the probability that the next call received by the office is received more than 10 minutes later. Explain your answer.
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    Write the question numbers in the left-hand margin. Additional page, if required.
    Write the question numbers in the left-hand margin.
OCR FS1 AS 2021 June Q1
1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\mathrm { Po } ( 120 )\).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
  3. State a necessary assumption for the validity of your calculation in part (b).