Probability distributions from selection

Derive the complete probability distribution for a random variable defined by a selection process (e.g., number of vowels in selected letters, sum of numbers on selected cards).

5 questions · Moderate -0.0

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CAIE S1 2007 June Q7
10 marks Standard +0.3
7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.
  1. Find the probability that the three peppers are all different colours.
  2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
  3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).
CAIE S1 2004 November Q6
9 marks Standard +0.3
6 A box contains five balls numbered \(1,2,3,4,5\). Three balls are drawn randomly at the same time from the box.
  1. By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number. The random variable \(L\) denotes the largest of the three numbers drawn.
  2. Find the probability that \(L\) is 4 .
  3. Draw up a table to show the probability distribution of \(L\).
  4. Calculate the expectation and variance of \(L\).
CAIE S1 2016 November Q2
7 marks Moderate -0.8
2 Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses 3 items at random. The random variable \(X\) is the number of T-shirts chosen.
  1. Show that the probability that Noor chooses exactly one T-shirt is \(\frac { 27 } { 55 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2011 November Q3
6 marks Moderate -0.3
3 A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable \(X\) is the number of girls in the team.
  1. Draw up a probability distribution table for \(X\).
  2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 2 }\), calculate \(\operatorname { Var } ( X )\).
Edexcel S1 Q5
13 marks Standard +0.3
5. The letters of the word DISTRIBUTION are written on separate cards. The cards are then shuffled and the top three are turned over. Let the random variable \(V\) be the number of vowels that are turned over.
  1. Show that \(\mathrm { P } ( V = 1 ) = \frac { 21 } { 44 }\).
  2. Find the probability distribution of \(V\).
  3. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).