Independence test with conditional probability

Questions that provide P(A|B) or require calculating it, and test independence by checking if P(A|B) = P(A) or equivalent conditional probability relationships.

4 questions · Moderate -0.9

2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles
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OCR MEI S1 Q4
6 marks Moderate -0.8
4 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 2014 June Q3
6 marks Easy -1.2
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. [1]
  2. Find \(\mathrm{P}(L \cap R)\). [2]
  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. [3]
OCR MEI S1 Q3
6 marks Moderate -0.8
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 \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
  2. Find \(\text{P}(T \cap M)\). [2]
  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. [3]
OCR MEI Paper 2 Specimen Q7
4 marks Moderate -0.8
Two events \(A\) and \(B\) are such that \(\text{P}(A) = 0.6\), \(\text{P}(B) = 0.5\) and \(\text{P}(A \cup B) = 0.85\). Find \(\text{P}(A | B)\). [4]