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Sample spaces, outcomes and events The definition of probability The uniform probability measure §2.1 Probabilities, Events, and Equally Likely Outcomes Tom Lewis Fall Semester 2014 Sample spaces, outcomes and events The definition of probability Outline Sample spaces, outcomes and events The definition of probability The uniform probability measure The uniform probability measure Sample spaces, outcomes and events The definition of probability The uniform probability measure Definition An element of a sample space is called an outcome. A subset of a sample space is called an event. Problem Toss a coin three times in a row and record the result after each toss. Construct the event that... • exactly 2 heads are tossed; • exactly 3 heads are tossed; • at least 2 heads are tossed. Sample spaces, outcomes and events The definition of probability The uniform probability measure Problem 4 balls (labelled 1 through 4) are placed in an urn. An experiment consists of taking two balls from the urn (one at a time and without replacement). Construct the event that ... • the sum of the balls is 4; • the number on the second ball is greater than the number on the first. Sample spaces, outcomes and events The definition of probability The uniform probability measure Definition A certain experiment has a sample space X of the form X = {s1 , s2 , . . . , sn }. To each outcome si we associate a number wi called the weight or probability of the outcome. The weights must satisfy: • 0 6 wi 6 1 for all 1 6 i 6 n; • w1 + w2 + · · · + wn = 1 Here is some common notation: wi = P(si ). Sample spaces, outcomes and events The definition of probability The uniform probability measure Definition Let E ⊂ X be an event. We define the probability of the event E by the rule P(E ) = the sum of the weights of the outcomes that comprise E . Example Thus, if E = {s1 , s2 , s3 }, we would have P(E ) = w1 + w2 + w3 . Sample spaces, outcomes and events The definition of probability The uniform probability measure Problem Let X = {a, b, c, d , e} with P(a) = .1, P(b) = .1, P(c) = .2, P(d ) = .3 and P(e) = .3. Find P(E ) if... • E = {a, b, c}; • E = {d , e}; • E = {b, c}. Sample spaces, outcomes and events The definition of probability Theorem (The Complement Rule) Let E ⊂ X be an event. Then P(E ) + P(E 0 ) = 1. The uniform probability measure Sample spaces, outcomes and events The definition of probability The uniform probability measure Definition The uniform probability measure is the probability measure that assigns equal probability to every outcome in the sample space. Problem Let X = {a, b, c, d , e, f , g , h, i, j} be the sample space of a probability experiment. • How much weight does a uniform probability measure assign to each outcome? • Under a uniform measure, what is the probability of the event E = {b, d , f , g }? Sample spaces, outcomes and events The definition of probability The uniform probability measure Uniform probability measures Let X be a uniform probability space and let E ⊂ X . Then P(E ) = n(E ) n(X ) In other words, the probability of an event is the ratio of the number of outcomes in E to the number of outcomes in X . Sample spaces, outcomes and events The definition of probability The uniform probability measure Problem Toss three coins. What is the probability of tossing at least 2 heads? What is the probability of tossing exactly 2 heads? Sample spaces, outcomes and events The definition of probability The uniform probability measure Problem Toss a red and a green die. What is the probability that the dice match? What is the probability that the dice sum to 7? What is the probability that the dice sum to 4?