Independence in contingency tables

A question is this type if and only if it provides a two-way table of frequencies and asks to test whether two categorical variables (e.g., college and sport preference, gender and instrument) are independent.

5 questions

CAIE S1 2020 June Q2
2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}SoccerHockeyTotal
Amos543286
Benn8472156
Canton225678
Devar12060180
Total280220500
  1. Find the probability that a randomly chosen student is at Canton college and prefers hockey.
  2. Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.
  3. One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.
CAIE S1 2021 March Q7
7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table.
SwimmingCyclingRunning
Female1045066
Male315792
A student is chosen at random.
    1. Find the probability that the student prefers swimming.
    2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
    1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
    2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2016 June Q1
1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .
  1. Copy and complete the following table to show the number of adults in each category.
    Wears spectaclesDoes not wear spectaclesTotal
    Right-handed
    Not right-handed
    Total30
    An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
  2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.
CAIE S1 2019 November Q1
1 There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table.
GuitarPianoFlute
Female students623543
Male students784042
  1. Find the probability that a randomly chosen student at the college is a male who does not play the piano.
  2. Determine whether the events 'a randomly chosen student is male' and 'a randomly chosen student does not play the piano' are independent, justifying your answer.
AQA Paper 3 2018 June Q14
14 A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students:
12 study physics
8 study geography
4 study geography and physics
14
  1. A student is chosen at random from the class.
    Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent.
    14
  2. It is known that for the whole college:
    the probability of a student studying mathematics is \(\frac { 1 } { 5 }\)
    the probability of a student studying biology is \(\frac { 1 } { 6 }\)
    the probability of a student studying biology given that they study mathematics is \(\frac { 3 } { 8 }\)
    Calculate the probability that a student studies mathematics or biology or both.