05 Introduction to Probability Part 1

Two elementary outcomes are necessarily mutually

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Two elementary outcomes are necessarily mutually exclusive Examples: Rolling a three in a single die toss A circuit board passes a quality test Picking an ace of spades Event: Set of elementary outcomes of interest Examples: One or more heads in a two coin toss The sum of dice is greater than seven 8
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Basic Elements of Probability Theory Sample Space (S) Set of all possible elementary outcomes of an experiment (mutually exclusive and collectively exhaustive) Example: Sample space of a single die toss Example: Drawing a marble from a box An experiment consists of drawing one marble from a box that contains a mixture of red, yellow and green marbles. Sample space: Suppose in this experiment, you are interested in the number of red marbles drawn. Is there another relevant sample space? S={red, yellow, green} Yes. S={0,1} 9
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Drawing two marbles from a box Suppose a first marble is drawn, recorded, and replaced, and a second marble is drawn, recorded and replaced List a relevant sample space if you are interested in the color combos List a relevant sample space if you are interested in the number of reds S={rr, rg, ry, gr, gg, gy, yr, yg, yy} S={0, 1, 2} 10
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Visualizing Events Contingency tables : Tree diagrams: 11 Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Red Card Black C ard Not an Ace Ace Ace Not an Ace 2 24 2 24
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Assigning probabilities to individual outcomes Probability Space : A sample space with probabilities assigned to each elementary outcome Classical Probability Assessment All outcomes equally likely to occ ur Relative Frequency of Occurrence Based on actual observations Subjective Probability Assessment An opinion or judgment by a decision maker about the likelihood of an event based on their expertise 12 P(Ei) = Number of ways Ei can occur Total number of experimental outcomes Relative Freq. of Ei = Number of times Ei occurs N
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Assigning probabilities to individual outcomes Probability Space : A sample space with probabilities assigned to each elementary outcome according to the following rules Rule 1 : Probability of any elementary outcome must fall in the interval from zero to one Rule 2 : Probabilities for all elementary outcomes in an experiment (the sample space) must sum to one 13 outcome elementary each for 1 ) ( 0 i e P = S in outcomes elementary all 1 ) ( i e P
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Example: Drawing two marbles How many outcomes? S ={rr, rg, ry, gr, gg, gy, yr, yg, yy} Each of 9 outcomes is equally likely Each outcome assigned probability = 1/9 1/3 x 1/3 =1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 Second Draw First Draw red green yellow red green yellow 14
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Assigning probabilities to events For any event A P(A) = sum of the probabilities for the elementary outcomes in A If A = {a1,a2,a3} P(A) = Example: Drawing marbles A = event that at least one red is drawn B = event that two of the same color are drawn 1.
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