Multi-item selection from population

Questions involving selecting three or more items from a finite population without replacement, calculating probabilities of various compositions in the selection.

5 questions · Moderate -0.5

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Edexcel S1 2002 November Q2
7 marks Moderate -0.8
There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both. Three students are selected at random, without replacement. Find the probability that
  1. all three students are studying only arts subjects, [4]
  2. exactly one of the three students is studying only science subjects. [3]
Edexcel S1 Q6
17 marks Standard +0.3
Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac{1}{3}\).
  1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r(r - 1) = 420.$$ [4 marks]
  2. Hence or otherwise find the value of \(r\). [3 marks]
  3. Find the probability that, when three cards are drawn at random from the pack,
    1. at least two are red, [6 marks]
    2. the first one is red given that at least two are red. [4 marks]
OCR S1 2013 June Q6
7 marks Easy -1.3
The diagram shows five cards, each with a letter on it. \includegraphics{figure_6} The letters A and E are vowels; the letters B, C and D are consonants.
  1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them. [2]
  2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it. [3]
  3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it. [2]
OCR S1 2013 June Q8
7 marks Moderate -0.3
  1. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours. [3]
  2. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54. Find the number of red discs that were in the second bag at the start. [4]
AQA AS Paper 2 Specimen Q15
2 marks Moderate -0.3
A school took 225 children on a trip to a theme park. After the trip the children had to write about their favourite ride at the park from a choice of three. The table shows the number of children who wrote about each ride.
Ride written about
The DropThe BeanstalkThe GiantTotal
Year 724452392
Year 836172275
Year 920132558
Total807570225
Three children were randomly selected from those who went on the trip. Calculate the probability that one wrote about 'The Drop', one wrote about 'The Beanstalk' and one wrote about The Giant'. [2 marks]