Independence test requiring preliminary calculations

Questions where P(A), P(B), or P(A∩B) must first be calculated from given information (such as P(A∪B), Venn diagram data, or other relationships) before testing independence.

11 questions

OCR MEI S1 2008 June Q2
2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.
OCR MEI S1 Q5
5 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.
  • \(W\) is the event that Alan has to wait at the level crossing.
  • \(L\) is the event that Alan is late for work.
You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).
  1. Calculate \(\mathrm { P } ( L \cap W )\).
  2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.
OCR MEI S1 Q3
1 marks
3 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults. [1]
OCR MEI S1 2012 January Q4
4 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.
  • \(T\) is the event that this person likes tomato soup.
  • \(M\) is the event that this person likes mushroom soup.
You are given that \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent.
  2. Find \(\mathrm { P } ( T \cap M )\).
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2014 June Q3
3 Each weekday, Marta travels to school by bus. Sometimes she arrives late.
  • \(L\) is the event that Marta arrives late.
  • \(R\) is the event that it is raining.
You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent.
  2. Find \(\mathrm { P } ( L \cap R )\).
  3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2009 January Q5
5 Each day Anna drives to work.
  • \(R\) is the event that it is raining.
  • \(L\) is the event that Anna arrives at work late.
You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).
  1. Determine whether the events \(R\) and \(L\) are independent.
  2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.
Edexcel S1 2006 January Q6
6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$
  1. Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\).
  2. Write down the value of \(\mathrm { P } ( A )\) and the value of \(\mathrm { P } ( B )\).
  3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
  4. Determine whether or not \(A\) and \(B\) are independent.
Edexcel S1 Q5
  1. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A \cap B ) = 0.24 , \mathrm { P } ( A \cup B ) = 0.88\) and \(\mathrm { P } ( B ) = 0.52\).
    1. Find \(\mathrm { P } ( A )\).
    2. Determine, with reasons, whether \(A\) and \(B\) are
      1. mutually exclusive,
      2. independent.
    3. Find \(\mathrm { P } ( B \mid A )\).
    4. Find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).
    5. The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours.
      Use this model to calculate
    6. the probability that a person chosen at random took between 5 and 9 hours to complete the task,
    7. the range, symmetrical about the mean, within which \(80 \%\) of the people's times lie.
      (5 marks)
      It is found that, in fact, \(80 \%\) of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.
    8. Find the standard deviation of the times in the modified model.
    9. The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y \mathrm {~km}\) per litre, on a long journey.
    Car\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
    \(x\)0.951.201.371.762.252.502.875
    \(y\)21.317.215.519.114.711.49.0
    \(\sum x = 12 \cdot 905 , \sum x ^ { 2 } = 26 \cdot 8951 , \sum y = 108 \cdot 2 , \sum y ^ { 2 } = 1781 \cdot 64 , \sum x y = 183 \cdot 176\).
  2. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = a y + b\).
  3. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value.
  4. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be.
  5. Comment on the use of the line to find values of \(x\) as \(y\) gets very small.
Edexcel S1 Q2
2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),
  1. determine whether \(A\) and \(B\) are independent events.
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( A \mid C )\).
SPS SPS SM Mechanics 2021 January Q3
3. The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\)
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  1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\) Given that \(\mathrm { P } ( B ) = 0.4\)
  2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
  3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
  4. find
    1. the value of \(r\)
    2. the value of \(s\)
SPS SPS SM 2021 February Q3
3. The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\)
\includegraphics[max width=\textwidth, alt={}, center]{64d3256a-9007-4e8a-86d4-8375c006a4ce-05_524_897_351_625}
  1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\) Given that \(\mathrm { P } ( B ) = 0.4\)
  2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
  3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
  4. find
    1. the value of \(r\)
    2. the value of \(s\)