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Intersection and conditional probability from a union of two events, and a test of whether the events are independent

Intersection and conditional probability from a union of two events, and a test of whether the events are independent

6 marks

For two events \(C\) and \(D\), \(\mathrm{P}(C)=\dfrac25\), \(\mathrm{P}(D)=\dfrac13\) and \(\mathrm{P}(C\cup D)=\dfrac{8}{15}\).

  1. Find \(\mathrm{P}(C\cap D)\).
  2. Find \(\mathrm{P}(C\mid D)\).
  3. Determine whether \(C\) and \(D\) are independent events. Justify your answer.

[6]

Solution:

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Answer:(a) \(\dfrac15\) (b) \(\dfrac35\) (c) not independent, since \(\mathrm{P}(C\mid D)=\dfrac35\ne\mathrm{P}(C)=\dfrac25\)

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