Unit11 - Unit 11 Probability and Calculus I Discrete...

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Unit 11 Probability and Calculus I Discrete Distributions Terminology: Each problem will concern an experiment , which can result in any one of a number of diFerent outcomes . The set consisting of all possible outcomes of an experiment is called the sample space, S , for the experiment. ±or any set A , we use | A | to denote the number of elements in A (the size of A ). Thus | S | is the number of diFerent possible outcomes of the experiment. Any collection of outcomes (i.e. any subset of the sample space) is called an event . Associated with each event is a number between 0 and 1 called its probability , which measures the likelihood of the event occuring when the experiment is performed. Notice: All of the outcomes in a sample space must be de²ned so as to be mutually exclusive, i.e., only one can occur at a time. As well, the sample space must include all of the possible outcomes. So any time that an exper- iment is performed, one and only one element of S is observed (i.e., occurs). Theorem 11.1. Let S be the sample space for some experiment and E be any event deFned on that sample space. If all outcomes of the experiment are equally likely to occur, then the probability associated with E , denoted prob ( E ) is given by prob ( E ) = | E | | S | i.e., prob ( E ) = number of outcomes in E ÷ number of outcomes in S . 1
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Example 1 . For the experiment ‘roll a fair die’, ±nd the probabilities of: A , the event that an even number is rolled, B , the event that an odd number is rolled, and C , the event that either a 1 or a 2 is rolled. Solution: If an experiment consists of rolling a fair die, then the possible outcomes are the numbers which may be rolled. We see that S = { 1 , 2 , 3 , 4 , 5 , 6 } with | S | = 6 and we have A: roll an even number A = { 2 , 4 , 6 } so | A | = 3 B: roll an odd number B = { 1 , 3 , 5 } so | B | = 3 C: roll a 1 or a 2 C = { 1 , 2 } so | C | = 2 Since each number is equally likely to come up when we roll a fair die, we have: prob ( A ) = | A | | S | = 3 6 = 1 2 prob ( B ) = | B | | S | = 3 6 = 1 2 prob ( C ) = | C | | S | = 2 6 = 1 3 When the various outcomes of an experiment are not equally likely to oc- cur, then we ±nd the probability of an event by adding up the probabilities of the various outcomes contained in the event. Notice: This is e²ectively what we have done for equally likely outcomes, too. Example 2 . An urn contains 1 yellow ball, 2 red balls, 3 blue balls and 4 green balls. One ball is drawn from the urn at random. Find the probability that either a yellow or a blue ball is drawn. Solution: We can de±ne the outcomes of this experiment as the colour of the ball drawn. Letting Y , R , B and G denote yellow, red, blue and green, respectively, we have S = { Y,R,B,G } .
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This note was uploaded on 11/03/2009 for the course MATH 0110A taught by Professor Olds,vicky during the Fall '09 term at UWO.

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Unit11 - Unit 11 Probability and Calculus I Discrete...

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