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Lecture 02 - 1 Lecture Plan Experiments Outcomes and Events...

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1 Lecture Plan Experiments, Outcomes and Events The Axioms of Probability Axiom consequences Finite Sample Space. 2 Experiments, Outcomes, Sample Space, and Events Example: Experiment: Toss a coin three times. Outcomes: The possible outcomes are hhh, hht, hth, htt, thh, tht, tth, ttt . Sample space: The set of all outcomes: S = { hhh, hht, hth, htt, thh, tht, tth, ttt } Events: Are subsets of S to which we will assign probabilities. Examples of events: A at least one head: A = { hhh, hht, hth, htt, thh, tht, tth } , B the first two tosses are heads: B = { tth, ttt } An event A is said to have occurred if any outcome ω A occurs when an experiment is conducted. Suppose an experiment is done with outcome ω = tth then all events (subsets of S ) containing ω occur. 2.1 The Axioms of Probability See section 2.4 pages 69-74. In probability we are interested in assigning numbers in [0 , 1] to certain subsets of S (called events). The following axioms allow us to do this in a consistent way. A1 If A S is an event then P ( A ) 0 . A2 P ( S ) = 1 A3 If A 1 , A 2 , . . . are mutually exclusive events then P ( n i =1 A i ) = n X i =1 P ( A i ) for all n = 1 , 2 , . . . , . 1
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Note: In order for a set of outcomes to be an event we need to be able to assign a probability to the set.
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