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Definitions Experiment – a process by which an observation ( or measurement ) is observed Sample Space (S)- The set of all possible outcomes (or results) of an experiment Event (E) – a collection of outcomes i.e E S 1 Example Experiment : Toss a balanced die once and observe its uppermost face Sample Space =S={1,2,3,4,5,6} Events: 1.observe a even number E= { 2,4,6} 2. observe a number less than or equal to 4 F= { 1,2,3,4} 2 Probability Given a event (E) , we would like to assign it a number, P(E) P(E) is called the probability of E (likelihood that E will occur) 0 P( E ) 1 3 Practical Interpretation The fraction of times that E happens out of a huge number of trials of the same experiment will be close to P(E) Types of Probabilities 4 Theoretical Empirical Theoretical Probabilities Used if the outcomes of an experiment are equally likely to occur If E is an Event number of outcomes in event E P( E ) number of outcomes in sample space 5 Example Toss a balanced die once and observe its uppermost face S={1,2,3,4,5,6} Let G=“observe a number divisible by 3” G={3,6} Then P(G)=2/6=1/3 6 Empirical Probabilities Used when theoretical probabilities cannot be used The experiment is repeated large number of times If E is an Event number of times E happens P( E ) number of trials 7 Example The freshman class at ABC college - 770 students - 485 identified themselves as “smokers” Compute the empirical probability that a randomly selected freshman student from this class is not a smoker 8 Example-contd. 9 E= event that a randomly chosen student from this class is not a smoker P(E)= 285/770=0.37 Properties I 1. 0 P ( E ) 1 P( E ) 1 2. If E is certain to happen 3. If E and F cannot both happen P( E or F ) P( E ) P( F ) 4. 10 P( S ) 1 Union Def. The union of two sets, E and F, is the set of outcomes in E or F . Example: E= { 2,4,6} F= { 1,2,3,4} E F {1, 2, 3, 4, 6} 11 Intersection Def. The intersection of two sets, E and F, is the set of outcomes in E and F . Example: E= { 2,4,6} F= { 1,2,3,4} E F {2 ,4} 12 Mutually Exclusive Def. Two events, E and F, are mutually exclusive if they have no outcomes in common, i.e. . E F If E and F are mutually exclusive, then P( E F ) P( E ) P( F ) 13 This property can be extended to more than two events. For any two events, E and F, P( E F ) P( E ) P( F ) P( E F ) 14 Complement of an Event Def. The complement of an event, E, is the event that E does not happen . Example: S={1,2,3,4,5,6} E= { 2,4,6} E {1 ,3 ,5} C Does E and E have common outcomes? C 15 Since the two events are Mutually Exclusive P( E E C ) P( E ) P( E C ) P( S ) P( E ) P( E C ) 1 P( E ) P( E ) C P( E C ) 1 P( E ) 16 P( E C ) 1 P( E ) 1 1 2 1 2 17 18 Assign probability to each outcome Each probability must be between 0 and 1 The sum of the probabilities must be equal to 1 If the outcomes of an experiment are all equally likely, then the probability of each outcome is given by 1 ,where n is the number of possible outcomes n DeMorgan’s Laws P( E C F C ) P(( E F )C ) 1 P( E F ) P( E C F C ) P(( E F )C ) 1 P( E F ) 19 Project Focus • How can probability help us with the decision on whether or not to attempt a loan work out? Events: S- an attempted work out is successful F- an attempted work out fails Goal: P(S) – Probability of S or fraction of past work out arrangements which were successful P(F) - Probability of F or fraction of past work out arrangements which were unsuccessful? 20 Using “Countif” function in Excel 1. 2. 21 Counts the number of cells within a given range that meets the given criteria Fields for the function Range Criteria Project Focus – Basic Probability Counting Fractions Number of Number of Fraction of Fraction of Successes Failures Successes Failures 3,818 4,408 0.464138099 0.535861901 22 Estimated Probabilities P (S ) 0.464 P (F ) 0.536 More on Events S & F F is the complement of S Recall: P( S C ) 1 P( S ) P( F ) 1 P( S ) 23