Single scaled time period

Questions that require scaling the Poisson parameter to a single different time period and calculating probabilities for that period only.

4 questions · Moderate -0.5

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling
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OCR S2 2014 June Q4
7 marks Easy -1.2
4 A zoologist investigates the number of snakes found in a given region of land. The zoologist intends to use a Poisson distribution to model the number of snakes.
[0pt]
  1. One condition for a Poisson distribution to be valid is that snakes must occur at constant average rate. State another condition needed for a Poisson distribution to be valid. [1] Assume now that the number of snakes found in 1 acre of a region can be modelled by the distribution Po(4).
    [0pt]
  2. Find the probability that, in 1 acre of the region, at least 6 snakes are found. [2]
    [0pt]
  3. Find the probability that, in 0.77 acres of the region, the number of snakes found is either 2 or 3. [4]
Edexcel S2 2021 June Q2
15 marks Standard +0.3
  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.
OCR MEI Further Statistics A AS Specimen Q1
6 marks Moderate -0.8
1 The number of failures of a machine each week at a factory is modelled by a Poisson distribution with mean 0.45.
  1. Write down the variance of the distribution.
  2. Find the probability that there are exactly 2 failures in a week.
  3. State a distribution which can be used to model the number of failures in a period of 4 weeks.
  4. Find the probability that there are at least 2 failures in a period of 4 weeks.
CAIE S2 2020 Specimen Q3
5 marks Moderate -0.3
The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.
  1. Find the probability exactly 4 calls in a 10 minute period. [2]
  2. Find the probability at least 3 calls in a 3 minute period. [3]