Consecutive non-overlapping periods

Questions specifically about consecutive or successive time periods where the temporal ordering or sequence matters (e.g., first half vs second half, successive minutes).

6 questions

CAIE S2 2008 November Q6
6 In their football matches, Rovers score goals independently and at random times. Their average rate of scoring is 2.3 goals per match.
  1. State the expected number of goals that Rovers will score in the first half of a match.
  2. Find the probability that Rovers will not score any goals in the first half of a match but will score one or more goals in the second half of the match.
  3. Football matches last for 90 minutes. In a particular match, Rovers score one goal in the first 30 minutes. Find the probability that they will score at least one further goal in the remaining 60 minutes. Independently of the number of goals scored by Rovers, the number of goals scored per football match by United has a Poisson distribution with mean 1.8.
  4. Find the probability that a total of at least 3 goals will be scored in a particular match when Rovers play United.
OCR MEI S2 2007 June Q3
3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.
  1. Find the probability of
    (A) exactly one call in a 5 -minute period,
    (B) at least 6 calls in a 5 -minute period.
  2. Find the probability of
    (A) exactly one call in a 1 -minute period,
    (B) exactly one call in each of five successive 1-minute periods.
  3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive
    • at a uniform average rate,
    • independently of each other.
    • Comment briefly on the validity of each of these assumptions if the office is
      (A) the enquiry department of a bank,
      (B) a police emergency control room.
Edexcel S2 2018 January Q5
5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)
  1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
  2. Find the largest possible value of \(k\) In a randomly selected month,
  3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
  4. Find the probability that at least 10 packages were lost in each of these two months.
  5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.
Edexcel S2 2006 January Q2
2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.
  1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
  2. there will be 2 accidents in the same week,
  3. there is at least one accident per week for 3 consecutive weeks,
  4. there are more than 4 accidents in a 2 week period.
AQA S2 2006 January Q1
1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .
    1. Calculate \(\mathrm { P } ( X = 2 )\).
    2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
    1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
    2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
  3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.
AQA S2 2005 June Q1
1 The number of cars, \(X\), passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.
  1. Calculate \(\mathrm { P } ( X = 2 )\).
    1. Write down the distribution of \(Y\), the number of cars passing along this road in a 5-minute interval.
    2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.