Basic ProbabilitySetsSubsetsSample SpaceEvents (E)Event, con’tExerciseAnother ExerciseAnother Exercise (cont)Probability of an EventHow Probabilities are AssignedProperties of ProbabilitiesProbability, con’tVenn DiagramsSlide 15ComplementsUnions of setsIntersection of SetsProbabilitiesMutually ExclusiveSlide 21ReviewReview, con’tDe Morgan’s LawsAnswer the following questions:Basic ProbabilitySets, SubsetsSample SpaceEvent, EProbability of an Event, P(E)How Probabilities are assignedProperties of ProbabilitiesSetsA set is simply any collection of objectsA set may be finite or infiniteA set with nothing in it is called the empty set (null or void set) and is denoted, { } or øTwo sets are equal if they have exactly the same elementsSubsetsIf A={1,2,3} is a set then subsets of A include the sets:{ },{1},{2},{3},{1,2}, {1,3},{2,3},{1,2,3}If B is one of the subsets of A then we can say that B ASample SpaceThe set, S, of all distinct possible outcomes of an experiment is called a sample spaceSuppose we are rolling a die, what is the sample space, S?Suppose we toss a coin twice recording the outcome each time, what is the sample space, S?Events (E)An event is any collection of outcomes of a probability experimentSuppose we are flipping a coin-what are the events that may occur?Suppose we are rolling a die, what are the events that may occur?What if we flip the coin twice, what are the events that may occur?Event, con’tSince an event is a collection of outcomes, we can say that ESWhat does E mean?What does S stand for?What does ES mean?An event, E is a subset of a sample space, S.ExerciseDetermine the sample space for the experiment: flipping a coin three timesWrite three events that correspond to this experimentWhat is P(at least 3 heads)?What is P(at least 3 tails)?What is P(no heads)?What is P(no more than 1 head)?Another ExerciseSuppose as part of a survey on popular music two students are asked whether they like a certain CD, dislike it, or don’t care. What is the sample space?What are the chances that the first student likes the CD?What are the chances that the second student likes the CD?Another Exercise (cont)How about each of the students either liking or disliking the CD?How about one student liking the CD while the other doesn’t care?What about the second student liking the CD AND one of the students likes the CD while the other doesn’t care?Probability of an Event Given an event, we would assign it a number, P(E) called the probability of EThis number indicates the likelihood that the event will occur.We can find this number by setting up a ratio:number of ways event can occurtotal number of outcomesHow Probabilities are AssignedInitial Probabilities are usually assigned either:Empirically-when an experiment is repeated a large number of times and you observe the fraction of times E occursBy AuthorityBy Common AgreementProperties of ProbabilitiesProbabilities must satisfy the following properties:i. For any event, E, 0P(E)1 ii. If E is certain to happen then P(E)=1iii. If E and F are events where E and F cannot happen at the same time, then P(E or F)=P(E)+P(F)Probability, con’tA set of possible outcomes of an experiment is a sample space, S.P(S)=?Venn DiagramsThe Venn Diagram is made up of two or more overlapping circles or sets. It is often used in mathematics to show relationships between sets.Venn DiagramsHere is the Venn Diagram associated with the set A.ComplementsA complement of A is everything that is in the universal set, U, but not in the set A.The complement is the event that A does not happen.The complement is denoted, Ac.Here is the complement of set A.Unions of setsThe union of sets A and B is the set of all items that are either in A or B.We express union, ABIn math, the word “or” also includes members of both A and B.Intersection of SetsThe intersection of sets A and B is the set of all items that are in both A and B.We express intersection, AB.ProbabilitiesFor any events E and F:( ) ( ) ( ) ( )( ) 1 ( )cP E F P E P F P E FP E P E Mutually ExclusiveTwo events are mutually exclusive if AB=.If A and B are mutually exclusive, then A B ( ) ( )P A B P A P B Mutually ExclusiveIf no two events E1,E2. . . . En can happen at the same time, then 1 2 1 2..... ( ) ( ) ..... ( )n nP E E E P E P E P E ReviewUnion of two sets, AB, is all items that are in set A OR set BIntersection of two sets, AB, is all items that are in both A AND BA complement of event A, Ac, is everything that is in the universal set but not in the set ATwo events are mutually exclusive if AB=Review, con’tFor any events, E and F:P(EF)=P(E)+P(F)-P(EF)We can use this equation to solve for P(EF), P(E), P(F), or P(EF)However if E and F are mutually exclusive then: P(EF)=P(E)+P(F)P(Ec)=1-P(E)De Morgan’s LawsIf A and B are any two sets: cc ccc cA B A BA B A B Answer the following questions:What is:(AAc) = ?(AAc) = ?If E and F are mutually exclusive, what is (E F) =
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