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

Edexcel Paper 3 2024 June Q1
  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 S2 Q5
5. As part of a business studies project, 8 groups of students are each randomly allocated 10 different shares from a listing of over 300 share prices in a newspaper. Each group has to follow the changes in the price of their shares over a 3-month period. At the end of the 3 months, \(35 \%\) of all the shares in the listing have increased in price and the rest have decreased.
  1. Find the probability that, for the 10 shares of one group,
    1. exactly 6 have gone up in price,
    2. more than 5 have gone down in price.
  2. Using a suitable approximation, find the probability that of the 80 shares allocated in total to the groups, more than 55 will have decreased in value.
Edexcel Paper 3 Specimen Q5
  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 \(55 \%\)