Calculate single probability using Poisson approximation

Questions that require calculating one specific probability value (e.g., P(X<3), P(X=k), P(X≥a)) using Poisson approximation to the binomial, typically with justification of the approximation.

9 questions

CAIE S2 2011 June Q1
1 On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable \(X\) denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.
CAIE S2 2024 November Q1
1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\).
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CAIE S2 2014 June Q1
1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.
OCR S2 2007 June Q2
2 It is given that on average one car in forty is yellow. Using a suitable approximation, find the probability that, in a random sample of 130 cars, exactly 4 are yellow.
Edexcel S2 2024 June Q5
5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.
  1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
  2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
  3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
  4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.
Edexcel S2 2006 January Q4
4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).
Edexcel S2 2009 January Q5
  1. A factory produces components of which \(1 \%\) are defective. The components are packed in boxes of 10 . A box is selected at random.
    1. Find the probability that the box contains exactly one defective component.
    2. Find the probability that there are at least 2 defective components in the box.
    3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
    4. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
      1. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
      2. State the minimum number of visits required to obtain a significant result.
    6. State an assumption that has been made about the visits to the server.
    In a random two minute period on a Saturday the web server is visited 20 times.
  2. Using a suitable approximation, test at the \(10 \%\) level of significance, whether or not the rate of visits is greater on a Saturday.
Edexcel S2 2009 June Q1
  1. A bag contains a large number of counters of which \(15 \%\) are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.
    1. Find the probability of no more than 6 red counters in this sample.
    A second random sample of 30 counters is selected and the number of red counters is recorded.
  2. Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13.
Edexcel S2 2009 June Q5
  1. An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.
    1. In a document of 2000 words find the probability that the administrator makes 4 or more errors.
    The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
  2. Use a suitable approximation to calculate the probability that the report is accepted.