Probability : Combined events 1 Objectives When you have competed it you should * know the addition rule * know Venn diagram * know the mutually exclusive events Key terms: Addition rule, Mutually exclusive events, Venn diagram, Exhaustive events

Addition Rule If A and B are any two events of the same experiment then the probability of A or B or both occurring is given by P ( A or B or both ) = P(A) + P(B) - P(A and B) P ( A  B ) = P(A) + P(B) - P(AB) S Venn diagram A B AB

Example 1 Events A and B are such that P(A) = 0.6 , P(B) = 0.7 and P(AB) = 0.4. Find P(A or B or both) Solutiom P(A  B) = P(A) + P(B) - P(A  B) P(A  B) = 0.6 + 0.7 - 0.4 = 0.9 A  B1 = 0.2 A1  B = 0.3 A  B = 0.4

Mutually Exclusive Events If an event A can occur or an event B can occur but not both A and B can occur, then the two events A and B are said to be mutually exclusive. P(AB) = 0 P(A  B) = P(A) + P(B) B A

Example 2 In a race the probability that Martin wins is 0.3, the probability that Ali wins is 0.25 and the probability that Chun wins is 0.2. Find probability that (a) Martin or Chun wins (b) neither Chun nor Ali wins . Solution P( Martin or Chun ) = P(Martin) + P(Chun) P( Martin or Chun ) = 0.30 + 0.20 = 0.50 P(neither chun nor Ali) = 1 - P( Chun or Ali ) = 1 - ( 0.20 + 0.25 ) = 0.55

Example 3 Tests are carried out on two machines A and B to assess the likelihood that each machine will produce a faulty component. Faulty Not Faulty Machine A Machine B 8 4 5 3 A component is chosen at random from those tested. Find the probability that the component chosen (i) is from machine A (ii) is a faulty component from machine B Solution (i) 12/20 = 3/5 (ii) 5/20 = 1/20