State exact binomial distribution

A question is this type if and only if it asks to state or identify the exact binomial distribution B(n,p) for a given scenario, including parameter values.

3 questions

OCR MEI S2 2012 June Q2
2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.
  1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
  2. Explain why \(X\) has, approximately, a Poisson distribution.
  3. Use a Poisson approximating distribution to find
    (A) \(\mathrm { P } ( X = 1 )\),
    (B) \(\mathrm { P } ( X > 4 )\).
  4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
  5. Use this Normal approximating distribution to
    (A) find the probability that there are at least 90 births which have the mutation,
    ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.
OCR MEI S2 2016 June Q2
2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.
  1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
  2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
  3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
  4. Use this Poisson distribution to find the probability that there are
    (A) exactly two genes that mutate,
    (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
  5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.
Edexcel S2 2022 January Q5
5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045
  1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
  2. State one assumption necessary for this distribution to be a suitable model of this situation.
  3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
  4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
  5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.