Use a two-set population to calculate regions, conditional probability, mutual exclusivity and independence.

Use a two-set population to calculate regions, conditional probability, mutual exclusivity and independence.

12 marks
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In a class of 100 boys, 55 boys play football and 75 boys play rugby. Each boy must play at least one sport from football and rugby.

  1. [3]
    1. Find the number of boys who play both sports.
    2. Write down the number of boys who play only rugby.
  2. One boy is selected at random.[4]
    1. Find the probability that he plays only one sport.
    2. Given that the boy selected plays only one sport, find the probability that he plays rugby. Let A be the event that a boy plays football and B be the event that a boy plays rugby.
  3. Explain why A and B are not mutually exclusive.[2]
  4. Show that A and B are not independent.[3]

Solution:

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Answer:The events are not mutually exclusive because 30 boys play both. They are not independent because \(P(F\cap R)=0.30\ne0.55(0.75)\).

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