Repeated binomial experiments

Calculate probabilities involving multiple independent binomial experiments (e.g. probability that in 10 bags, at least 2 have a certain property).

3 questions · Standard +0.1

2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial
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OCR MEI S2 2006 January Q2
18 marks Standard +0.3
2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.
  1. For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
    (A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
    (B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
    (C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
  2. Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
    (A) State the exact distribution of the number of positive tests.
    (B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
  3. Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value \(h\). Find the value of \(h\) for which this athlete would test positive on average just once in 200 occasions.
Edexcel Paper 3 2024 June Q1
11 marks Standard +0.3
  1. Xian rolls a fair die 10 times.
The random variable \(X\) represents the number of times the die lands on a six.
  1. Using a suitable distribution for \(X\), find
    1. \(\mathrm { P } ( X = 3 )\)
    2. \(\mathrm { P } ( X < 3 )\) Xian repeats this experiment each day for 60 days and records the number of days when \(X = 3\)
  2. Find the probability that there were at least 12 days when \(X = 3\)
  3. Find an estimate for the total number of sixes that Xian will roll during these 60 days.
  4. Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.
Edexcel Paper 3 Specimen Q5
9 marks Moderate -0.3
  1. A company sells seeds and claims that \(55 \%\) of its pea seeds germinate.
    1. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.
    A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
  2. Assuming that the company's claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
  3. Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution. A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
  4. Assuming that the company's claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
  5. Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of \(55 \%\)