Finding maximum n for P(X=0) threshold

Questions requiring the largest n such that the probability of zero occurrences exceeds a given threshold (e.g., P(X=0) > 0.9 or P(X=0) ≥ 0.95).

9 questions · Standard +0.0

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities
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CAIE S2 2020 March Q4
7 marks Standard +0.3
4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
  1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
  2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).
CAIE S2 2003 November Q4
8 marks Standard +0.3
4 The number of emergency telephone calls to the electricity board office in a certain area in \(t\) minutes is known to follow a Poisson distribution with mean \(\frac { 1 } { 80 } t\).
  1. Find the probability that there will be at least 3 emergency telephone calls to the office in any 20-minute period.
  2. The probability that no emergency telephone call is made to the office in a period of \(k\) minutes is 0.9 . Find \(k\).
CAIE S2 2013 November Q4
8 marks Standard +0.8
4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.
  1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
  2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .
OCR S2 2009 June Q5
9 marks Moderate -0.8
5 In a large region of derelict land, bricks are found scattered in the earth.
  1. State two conditions needed for the number of bricks per cubic metre to be modelled by a Poisson distribution. Assume now that the number of bricks in 1 cubic metre of earth can be modelled by the distribution Po(3).
  2. Find the probability that the number of bricks in 4 cubic metres of earth is between 8 and 14 inclusive.
  3. Find the size of the largest volume of earth for which the probability that no bricks are found is at least 0.4.
Edexcel S2 2009 June Q8
13 marks Standard +0.3
8. A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
  1. Find the probability of exactly 4 faults in a 15 metre length of cloth.
  2. Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
  3. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
  4. Find the retailer's expected profit.
Edexcel S2 2017 June Q2
13 marks Standard +0.3
  1. A company receives telephone calls at random at a mean rate of 2.5 per hour.
    1. Find the probability that the company receives
      1. at least 4 telephone calls in the next hour,
      2. exactly 3 telephone calls in the next 15 minutes.
    2. Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2
    The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10
  2. Test at the 5\% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.
Edexcel FS1 AS 2023 June Q3
16 marks Standard +0.3
  1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.
The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907
  1. Find the value of \(A\) The tablecloths are sold in packets of 20
    A randomly selected packet is taken.
  2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
    The random variable \(X\) represents the number of these tablecloths that have no faults.
  3. Find
    1. \(\mathrm { E } ( X )\)
    2. \(\operatorname { Var } ( X )\)
  4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
    The piece of cloth has area 30 square metres and is found to have 6 faults.
  5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
  6. Write down the \(p\)-value for the test used in part (e).
WJEC Unit 2 Specimen Q3
7 marks Moderate -0.8
Cars arrive at random at a toll bridge at a mean rate of 15 per hour.
  1. Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval. [1]
  2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive. [3]
  3. Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3. [3]
SPS SPS ASFM Statistics 2021 May Q7
Moderate -0.3
A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
  1. Find the probability of exactly 4 faults in a 15 metre length of cloth. (2)
  2. Find the probability of more than 10 faults in 60 metres of cloth. (3)
A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
  1. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. (4)
The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.
  1. Find the retailer's expected profit. (4)