Probability of specific committee composition

Calculate the probability that a randomly selected committee or group has a specific composition (e.g., exactly k members from category A) using combinations and probability ratios.

10 questions · Moderate -0.4

5.01a Permutations and combinations: evaluate probabilities
Sort by: Default | Easiest first | Hardest first
CAIE S1 2010 November Q7
11 marks Standard +0.3
7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women.
  1. Find the number of possible committees that can be chosen.
  2. Find the probability that one particular man, Albert, and one particular woman, Tracey, are both on the committee.
  3. Find the number of possible committees that include either Albert or Tracey but not both.
  4. The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line for refreshments. Find the probability that the women are not next to each other in the queue.
CAIE S1 2015 November Q2
3 marks Moderate -0.8
2 A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.
CAIE S1 2018 November Q4
7 marks Moderate -0.3
4 Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.
  1. Find the probability that the group of 7 includes one particular boy.
  2. Find the probability that the group of 7 includes at least 2 girls.
OCR MEI S1 Q3
7 marks Moderate -0.3
3 At a dog show, three out of eleven dogs are to be selected for a national competition.
  1. Find the number of possible selections.
  2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.
OCR S1 2015 June Q6
8 marks Moderate -0.8
6
  1. The seven digits \(1,1,2,3,4,5,6\) are arranged in a random order in a line. Find the probability that they form the number 1452163.
  2. Three of the seven digits \(1,1,2,3,4,5,6\) are chosen at random, without regard to order.
    1. How many possible groups of three digits contain two 1s?
    2. How many possible groups of three digits contain exactly one 1?
    3. How many possible groups of three digits can be formed altogether?
OCR MEI S1 2013 January Q4
7 marks Moderate -0.8
4 At a dog show, three out of eleven dogs are to be selected for a national competition.
  1. Find the number of possible selections.
  2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.
OCR MEI S1 2015 June Q4
6 marks Moderate -0.8
4 A rugby team of 15 people is to be selected from a squad of 25 players.
  1. How many different teams are possible?
  2. In fact the team has to consist of 8 forwards and 7 backs. If 13 of the squad are forwards and the other 12 are backs, how many different teams are now possible?
  3. Find the probability that, if the team is selected at random from the squad of 25 players, it contains the correct numbers of forwards and backs.
OCR Further Statistics 2018 December Q3
7 marks Standard +0.8
3
  1. Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
  2. Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.
OCR S1 Q7
14 marks Moderate -0.3
7 A committee of 7 people is to be chosen at random from 18 volunteers.
  1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
  2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
  3. include exactly 5 people from Worcester,
  4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
  5. Jenny estimates that her chances of success are as follows.
    • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
    • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
    • John estimates that his chances of success are as follows.
    • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
    • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
    Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
  6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
  7. Describe briefly how the heights of the second group of plants compare with the first.
  8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
  9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
  10. The probability distribution of \(X\) is given in the table below.
    \(x\)2345
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)
    Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2014 June Q4
6 marks Moderate -0.8
There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.
  1. Find the probability that all 4 members of the team will be girls. [3]
  2. Find the probability that the team will contain at least one girl and at least one boy. [3]