Multiple periods with binomial structure

Questions that first use normal approximation for a Poisson distribution, then apply this to multiple independent periods using binomial probability (e.g., probability that x out of n weeks satisfy a condition).

11 questions

CAIE S2 2012 June Q7
7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.
  1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
  2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
  3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .
OCR S2 2009 January Q3
3 The number of incidents of radio interference per hour experienced by a certain listener is modelled by a random variable with distribution \(\operatorname { Po } ( 0.42 )\).
  1. Find the probability that the number of incidents of interference in one randomly chosen hour is
    (a) 0 ,
    (b) exactly 1 .
  2. Find the probability that the number of incidents in a randomly chosen 5-hour period is greater than 3.
  3. One hundred hours of listening are monitored and the numbers of 1 -hour periods in which 0,1 , \(2 , \ldots\) incidents of interference are experienced are noted. A bar chart is drawn to represent the results. Without any further calculations, sketch the shape that you would expect for the bar chart. (There is no need to use an exact numerical scale on the frequency axis.)
Edexcel S2 2014 January Q5
5. A school photocopier breaks down randomly at a rate of 15 times per year.
  1. Find the probability that there will be exactly 3 breakdowns in the next month.
  2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
  3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
  4. Find the probability that the head teacher will buy a new photocopier.
Edexcel S2 2021 January Q3
3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7
  1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
  2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
    1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
    2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
  4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
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Edexcel S2 2022 January Q1
1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528
  1. Find the value of \(x\)
  2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
  3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".
Edexcel S2 2014 June Q7
7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .
  1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
  2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
  3. Find the value of \(x\).
Edexcel S2 2022 October Q1
  1. Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.
    1. Find the probability that in 15 metres of her cloth there are
      1. less than 3 faults,
      2. at least 6 faults.
    Each roll contains 100 metres of Bhavna's cloth.
    She selects 15 rolls at random.
  2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
    Each piece of her cloth is 4 metres long.
    The cost to make each piece is \(\pounds 5.00\)
    She sells each piece of her cloth that contains no faults for \(\pounds 7.40\)
    She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
  3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.
Edexcel S2 2012 January Q5
  1. The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
    1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box.
    A retailer buys 2 boxes of components.
  2. Estimate the probability that there are at least 4 defective components in each box.
Edexcel S2 2002 June Q6
6. From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m .
  1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. The twine is usually sold in balls of length 100 m . A customer buys three balls of twine.
  2. Find the probability that only one of them will have fewer than 6 faults. As a special order a ball of twine containing 500 m is produced.
  3. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive.
    (6)
Edexcel S2 2018 June Q1
  1. In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9
    1. Find
      1. \(\mathrm { P } ( X > 5 )\)
      2. \(\mathrm { P } ( 4 \leqslant X < 10 )\)
    The length of a working day is 7 hours.
  2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
  3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.
Edexcel FS1 AS 2020 June Q1
  1. A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.
    1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day.
    The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
  2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
    1. find the expected number of days that the company receives more than 28 call-outs,
    2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
    3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.