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AP Statistics Section 6.3 B Conditional probability Slim considers himself a pretty good poker player, at least when he is the only one playing. He has been dealt 4 cards and wishes to know the probability that his 5th card will be an ace. Can we figure this probability? Not without knowing what the first four cards were Find P(5th card is an ace) if his first 4 cards are two 3s, a 7 and a jack 4 1 48 12 Find P(5th card is an ace) if his first 4 cards are two 3s, a 7 and an ace. 3 1 48 16 The probability we assign to an event can change if we know that some other event has occurred. When a probability is based on the knowledge of a previous event it is called conditional probability The notation for conditional P( A / B) This probability is _______. notation is read: probability of A given that B has already occurred. Example: Here is a table of grades awarded at a university by school. Grade Level School A B Below B Total Liberal Arts 2,142 1,890 2,268 6,300 Engineering 368 432 800 1,600 Health Services 882 630 388 2,100 3,392 2,952 3,656 10,000 Total Consider the events: E = grade comes from an Engineering course B = the grade is a B. P(B) 2952 369 10000 1250 P(B/E) 432 27 1600 100 Example: Here is a table of grades awarded at a university by school. Grade Level School A B Below B Total Liberal Arts 2,142 1,890 2,268 6,300 Engineering 368 432 800 1,600 Health Services 882 630 388 2,100 3,392 2,952 3,656 10,000 Total Consider the events: E = grade comes from an Engineering course B = the grade is a B. P(E ) 1600 4 10000 25 P(E/B) 432 6 2952 41 Example: Here is a table of grades awarded at a university by school. Grade Level School A B Below B Total Liberal Arts 2,142 1,890 2,268 6,300 Engineering 368 432 800 1,600 Health Services 882 630 388 2,100 3,392 2,952 3,656 10,000 Total Consider the events: E = grade comes from an Engineering course B = the grade is a B. 1600 2952 432 4120 103 10000 10000 250 432 2 10000 25 P( E B) P(E B) Note: In conditional probability the condition has the effect of reducing the size of the sample space (i.e. the denominator in the probability fraction) General Multiplication Rule for Any Two Events: P( A B) P( A) P( B / A) Example: Slim is still at the poker table. Slim sees 11 cards on the table. Of these, 4 are diamonds. What is the probability of Slim being dealt 2 diamonds from the deck? 9 8 72 9 41 40 1640 205 If we take the General Multiplication Rule above and divide both sides by P(A) we obtain P( B A) P( B / A) P( A) Example: Motor vehicles are classified as either light trucks or cars and as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were domestic and 55% were domestic light trucks. Let T be the event a vehicle is a light truck and D be the event it is domestic. Write each of the following in terms of events T and D and give the probability. a. The vehicle is a car P(T ) 1 P(T ) 1 .69 .31 c Example: Motor vehicles are classified as either light trucks or cars and as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were domestic and 55% were domestic light trucks. Let T be the event a vehicle is a light truck and D be the event it is domestic. Write each of the following in terms of events T and D and give the probability. b. The vehicle is an imported car P( D T ) 1 (.14 .55 .23) .08 c c Example: Motor vehicles are classified as either light trucks or cars and as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were domestic and 55% were domestic light trucks. Let T be the event a vehicle is a light truck and D be the event it is domestic. Write each of the following in terms of events T and D and give the probability. c. If a vehicle is a car, what is the probability that it is imported? c c P ( D T ) .08 8 c c P( D / T ) c .31 31 P(T ) Example: Motor vehicles are classified as either light trucks or cars and as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were domestic and 55% were domestic light trucks. Let T be the event a vehicle is a light truck and D be the event it is domestic. Write each of the following in terms of events T and D and give the probability. d. Are the events “vehicle is a car” and “vehicle is imported” independent? Does P(T c D c ) P(T c ) P( D c ) .08 .31 .22 .08 .0682 Not Independent Does P( D c / T c ) P( D c ) 8 .22 31 Not Independent