NSW Y11 Maths - Extension 1 Permutations and Combinations Applications In Probability

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Applications In Probability Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-A1.1: Permutations and combinations. This will require student to 

  • list and count the number of ways an event can occur
  • use the fundamental counting principle (also known as the multiplication principle)
  • use factorial notation to describe and determine the number of ways 𝑛 different items can be arranged in a line or a circle
  • solve simple problems and prove results using the pigeonhole principle (ACMSM006)
  • understand and use permutations to solve problems (ACMSM001)
  • solve problems involving permutations and restrictions with or without repeated objects (ACMSM004)
  • understand and use combinations to solve problems (ACMSM007)
  • solve practical problems involving permutations and combinations, including those involving simple probability situations

Ref: https://educationstandards.nsw.edu.au/

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Theory

\(P(E) = \dfrac{{{\rm{number of favourable outcomes}}}}{{{\rm{number of possible outcomes}}}}\)

Mutually exclusive events: \(P({\rm{A or B)}} = P({\rm{A)}} + P({\rm{B)}}\)

Not mutually exclusive events: \(P({\rm{A or B)}} = P({\rm{A)}} + P({\rm{B)}} - P({\rm{A and B)}}\)

Independent events: \(P({\rm{A and B)}} = P({\rm{A)}} \times P({\rm{B)}}\)

For example: There are 4 red, 2 black and 3 white balls in a bag. What is the probability of selecting 3 red balls?

Combinations of red balls \( = {}^4{C_3} = 4\)

Total combinations\( = {}^9{C_3} = 84\)

Probability (3 Red)\( = \dfrac{4}{{84}} = \dfrac{1}{{21}}\)