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P (E) = PROBABILITY – STUDY NOTES Number of favourable outcomes ----------------------------------Total number of outcomes - n(E) n(S) COMPLEMENTARY EVENTS If P(E) is the probability of E occurring, then P(Ẽ) is the probability of E not occurring. Ẽ is called the complementary event, and P(E) + P(Ẽ) = 1. P(Ẽ) = 1 - P(E) EXPERIMENTAL PROBABILITY The experimental probability of an event occurring is its relative frequency. Relative frequency of an event = Frequency of the event ------------------------Total frequency THE MULTIPLICATION PRINCIPLE FOR COUNTING ARRANGEMENTS 1. If A can be arranged in m ways and B in n ways, then A and B together can be arranged in m x n. 2. More generally, if A can be arranged in a ways, B can be arranged in b ways, C can be arranged in c ways, etc., then A, B, C, … together can be arranged in a x b x c … ways. COUNTING ARRANGEMENTS The number of ways in which n different items can be arranged is n! = n (n - 1) x (n - 1) x … x 2 x 1 The number of ways in which n different items can be arranged in r positions is n! = n (n - 1) x (n - 1) x … [r terms] Example - The number of ways 6 items can be arranged in 4 positions is 6x5x4x3 [4 terms] COUNTING UNORDERED SELECTIONS The number of unordered selections that can be made from n different items, when there are r positions = n x (n - 1) x (n - 2) x … --------------------------r! [r terms] Example, the number of unordered selections that can be made from 6 different items, when there are 4 positions = 6x5x4x3 -------------4x3x2x1 PROBABILITY TREE DIAGRAMS To calculate the probability of a particular outcome, multiply the probabilities listed on the branches of each stage of the outcome. To calculate the probability of an event with two or more outcomes, add their calculated probabilities together. For complementary events, P(‘at least one’) = 1 – P(‘none’) EXPECTATION If the probability of an event E is p and the experiment is conducted n times, the expected number of times E will occur is n x p Financial expectation is calculated by multiplying every possible financial outcome by its probability and adding the results together. Probability: Meaning of probability Expectation Ordered selections Unordered selections Tree diagrams Probability in testing Simulation Multiplication