Region probabilities in a two-event Venn diagram, a probability conditional on a complement and a test for independence

Region probabilities in a two-event Venn diagram, a probability conditional on a complement and a test for independence

6 marks

For two events \(A\) and \(B\), \(P(A)=0.5\), \(P(B)=0.7\) and \(P(A\cap B)=0.3\). These events are represented in the Venn diagram below, where \(p\), \(q\) and \(r\) are the probabilities of three of its regions.

  1. Write down the value of
    1. \(p\)
    2. \(q\)
    3. \(r\)[3]
  2. Find \(P(A\mid B')\).[2]
  3. Hence, or otherwise, show that \(A\) and \(B\) are not independent events.[1]

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Answer:(a)(i) \(0.2\) (a)(ii) \(0.4\) (a)(iii) \(0.1\) (b) \(\frac{2}{3}\) (c) Not Independent

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