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Probability of an intersection from a conditional probability, then a test for independence

Probability of an intersection from a conditional probability, then a test for independence

6 marks

For two events \(C\) and \(D\), it is known that \(\mathrm{P}(C')=\frac23\), \(\mathrm{P}(D\mid C)=\frac34\) and \(\mathrm{P}(C\cup D)=\frac56\).

  1. Find \(\mathrm{P}(C\cap D)\).[3]
  2. Show that the events \(C\) and \(D\) are independent.[3]

Solution:

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Answer:(a) \(\frac14\) (b) \(\mathrm{P}(C)\,\mathrm{P}(D)=\frac14=\mathrm{P}(C\cap D)\)

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