AGENDA Probability Sample spaces and events Counting Probability Conditional Probability Bayes Theorm Mathematical Expectation SAMPLE SPACES AND EVENTS Sample space Set of all possible experimental outcomes Discrete Finite number of elements Only certain points between to values Continuous Continuum of elements All points between to values SAMPLE SPACES AND EVENTS Mutually exclusive events No elements in common Unions Either Intersections Both Complements Not VENN DIAGRAMS VENN DIAGRAMS VENN DIAGRAMS COUNTING Determining the number of possible elements in a solution Tree Diagrams Multiplication of choices N1 * n2 *n3… Multiple choice test with 12 answers = 2^12=4096 possibilities Permutations Particular order… Combinations No particular order… PERMUTATIONS Particular order (A, B, C) is different from (B, C, A) Many, many elements in the solution set “Permute yourself to death” in search of a solution n objects r objects selected PERMUTATIONS COMBINATIONS No particular order - order does not matter Any solution set of elements is the same (A, B, C) is the same as (B, C, A) Fewer elements in a solution than permutations COMBINATIONS PROBABILITY n equally likely possibilities s successes Probability of success is s/n Probability examples Getting a 6 on a dice roll = 1/6 Drawing an ace in a deck of cards = 4/52 Elementary Probability Theorms… ELEMENTARY PROBABILTY THEORMS If A1, A2, …An are mutually exclusive events then P(A1 U A2 U…An) = P(A1) + P(A2) + …P(An) Addition rule for any events Complement Rule CONDITIONAL PROBABILITY What is the probability of event A given event B has occurred Example Probability of high fidelity =0.81 Probability of both high fidelity and high selectivity=0.18 Probability of high selectivity with high fidelity=0.18 / 0.81 BAYES THEORM Extension of Conditional Probability If B1, B2,…Bn are mutually exclusive events of which one must occur then the probability of Br given event A is: BAYES THEORM EXAMPLE Repairs Janet services 20%, incomplete repair in 1/20 Tom services 60%, incomplete repair in 1/10 Georgia services 15%, incomplete repair in 1/10 Peter services 5%, incomplete repair in 1/20 If a repair is incomplete what is the probability it was performed by Janet? =(0.2)*(.05) / ((0.6)*(0.1)+(0.15)*(0.1)+(0.05)*(0.05) =0.114 MATHEMATICAL EXPECTATION Mathematical Expectation For amounts a1, a2, …ak With probabilities p1, p2, …pk E=a1*p1+a2*p2+…ak*pk Example 1000 raffle tickets Winning prize is $500 E for a single ticket=1/1000 * $500=$0.50