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Union of two events, a test for independence and a conditional probability given a complement

Union of two events, a test for independence and a conditional probability given a complement

6 marks

A member of a sports club is chosen at random. \(E\) is the event that the member swims and \(F\) is the event that the member cycles. It is known that \(\mathrm{P}(E)=0.35\), \(\mathrm{P}(F)=0.4\) and \(\mathrm{P}(E\cap F)=0.2\).

  1. Find \(\mathrm{P}(E\cup F)\).[2]
  2. Show that the events \(E\) and \(F\) are dependent.[1]
  3. Find the probability that the chosen member swims, given that the member does not cycle.[3]

Solution:

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Answer:(a) \(0.55\) (b) \(\mathrm{P}(E)\,\mathrm{P}(F)=0.14\neq0.2\) (c) \(\dfrac14\)

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