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Counts in two overlapping activities, a conditional probability, and why the events are neither mutually exclusive nor independent

Counts in two overlapping activities, a conditional probability, and why the events are neither mutually exclusive nor independent

12 marks

A sports club has \(120\) members. Every member swims, cycles, or does both. Of the members, \(78\) swim and \(69\) cycle.

    1. Find the number of members who both swim and cycle.
    2. Write down the number of members who cycle but do not swim.[3]
  2. A member of the club is chosen at random.
    1. Find the probability that the member does exactly one of the two activities.
    2. The member chosen does exactly one of the two activities. Find the probability that this member cycles.[4]

Let \(S\) be the event that a member chosen at random swims, and \(C\) the event that the member cycles.

  1. Explain why \(S\) and \(C\) are not mutually exclusive.[2]
  2. Show that \(S\) and \(C\) are not independent.[3]

Solution:

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Answer:(a)(i) \(27\) (a)(ii) \(42\) (b)(i) \(\dfrac{31}{40}\) (b)(ii) \(\dfrac{14}{31}\) (c) \(27\) members do both, so \(\mathrm{P}(S\cap C)\ne0\) (d) \(\mathrm{P}(S\cap C)=0.225\ne\mathrm{P}(S)\,\mathrm{P}(C)=0.37375\)

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