MATH 105: Finite Mathematics 7-5: Independent Events Prof. Jonathan - PowerPoint PPT Presentation

Introduction to Indepencence Examples Conclusion MATH 105: Finite Mathematics 7-5: Independent Events Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

Introduction to Indepencence Examples Conclusion Outline Introduction to Indepencence 1 Examples 2 Conclusion 3

Introduction to Indepencence Examples Conclusion Outline Introduction to Indepencence 1 Examples 2 Conclusion 3

Introduction to Indepencence Examples Conclusion Conditional Probability In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick ( C ) or thin crust and extra cheese ( E ) or regular. You select a person at random. Use the results below to find Pr[ C ] and Pr[ C | E ]. Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[ C ] = 40 60 = 2 Pr[ C | E ] = 24 36 = 2 3 3

Introduction to Indepencence Examples Conclusion Conditional Probability In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick ( C ) or thin crust and extra cheese ( E ) or regular. You select a person at random. Use the results below to find Pr[ C ] and Pr[ C | E ]. Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[ C ] = 40 60 = 2 Pr[ C | E ] = 24 36 = 2 3 3

Introduction to Indepencence Examples Conclusion Conditional Probability In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick ( C ) or thin crust and extra cheese ( E ) or regular. You select a person at random. Use the results below to find Pr[ C ] and Pr[ C | E ]. Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[ C ] = 40 60 = 2 Pr[ C | E ] = 24 36 = 2 3 3

Introduction to Indepencence Examples Conclusion Conditional Probability In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick ( C ) or thin crust and extra cheese ( E ) or regular. You select a person at random. Use the results below to find Pr[ C ] and Pr[ C | E ]. Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[ C ] = 40 60 = 2 Pr[ C | E ] = 24 36 = 2 3 3

Introduction to Indepencence Examples Conclusion Independent Events It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the other. That is, Pr[ E | F ] = Pr[ E ] Tests for Independence Test for independence using the formula: Pr[ E ∩ F ] = Pr[ E ] · Pr[ F ] or, use a Venn Diagram and determine if the ratio of E ∩ F to F is the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion Independent Events It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the other. That is, Pr[ E | F ] = Pr[ E ] Tests for Independence Test for independence using the formula: Pr[ E ∩ F ] = Pr[ E ] · Pr[ F ] or, use a Venn Diagram and determine if the ratio of E ∩ F to F is the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion Independent Events It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the other. That is, Pr[ E | F ] = Pr[ E ] Tests for Independence Test for independence using the formula: Pr[ E ∩ F ] = Pr[ E ] · Pr[ F ] or, use a Venn Diagram and determine if the ratio of E ∩ F to F is the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion Outline Introduction to Indepencence 1 Examples 2 Conclusion 3

Introduction to Indepencence Examples Conclusion A Venn Diagram Example Example Let A and B be events in a sample space S such that Pr[ A ] = 4 16 , Pr[ B ] = 8 16 , and Pr[ A ∩ B ] = 2 16 . Are A and B independent? Pr[ A ∩ B ] = 2 16 = 1 8 Pr[ A ] · Pr[ B ] = 4 16 · 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion A Venn Diagram Example Example Let A and B be events in a sample space S such that Pr[ A ] = 4 16 , Pr[ B ] = 8 16 , and Pr[ A ∩ B ] = 2 16 . Are A and B independent? Pr[ A ∩ B ] = 2 16 = 1 A B 8 Pr[ A ] · Pr[ B ] = 4 16 · 8 16 = 1 2 2 6 16 16 16 8 6 16 Independent!

Introduction to Indepencence Examples Conclusion A Venn Diagram Example Example Let A and B be events in a sample space S such that Pr[ A ] = 4 16 , Pr[ B ] = 8 16 , and Pr[ A ∩ B ] = 2 16 . Are A and B independent? Pr[ A ∩ B ] = 2 16 = 1 A B 8 Pr[ A ] · Pr[ B ] = 4 16 · 8 16 = 1 2 2 6 16 16 16 8 6 16 Independent!

Introduction to Indepencence Examples Conclusion A Venn Diagram Example Example Let A and B be events in a sample space S such that Pr[ A ] = 4 16 , Pr[ B ] = 8 16 , and Pr[ A ∩ B ] = 2 16 . Are A and B independent? Pr[ A ∩ B ] = 2 16 = 1 A B 8 Pr[ A ] · Pr[ B ] = 4 16 · 8 16 = 1 2 2 6 16 16 16 8 6 16 Independent!

Introduction to Indepencence Examples Conclusion A Venn Diagram Example Example Let A and B be events in a sample space S such that Pr[ A ] = 4 16 , Pr[ B ] = 8 16 , and Pr[ A ∩ B ] = 2 16 . Are A and B independent? Pr[ A ∩ B ] = 2 16 = 1 A B 8 Pr[ A ] · Pr[ B ] = 4 16 · 8 16 = 1 2 2 6 16 16 16 8 6 16 Independent!

Introduction to Indepencence Examples Conclusion Independence and Tree Diagrams Example A fair coin is tossed twocie and events E and F are defined as: E : Heads on the first toss F : Tails on the second toss Are E and F independent? Use a tree diagram to find out. Example In a group of seeds, 1 3 of which should produce violets, the best germinateion that can be obtained is 60%. If one seed is planted, what is the probability it will grow a violet? Assume the events are independent. Solve both using a tree diagram and a Venn diagram.

Introduction to Indepencence Examples Conclusion Independence and Tree Diagrams Example A fair coin is tossed twocie and events E and F are defined as: E : Heads on the first toss F : Tails on the second toss Are E and F independent? Use a tree diagram to find out. Example In a group of seeds, 1 3 of which should produce violets, the best germinateion that can be obtained is 60%. If one seed is planted, what is the probability it will grow a violet? Assume the events are independent. Solve both using a tree diagram and a Venn diagram.

Introduction to Indepencence Examples Conclusion A Used Car Lot Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[ B ] = 8 16 = 1 Pr[ D ] = 6 16 = 3 2 8 Pr[ B ∩ D ] = 5 16 � = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion A Used Car Lot Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[ B ] = 8 16 = 1 Pr[ D ] = 6 16 = 3 2 8 Pr[ B ∩ D ] = 5 16 � = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion A Used Car Lot Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[ B ] = 8 16 = 1 Pr[ D ] = 6 16 = 3 2 8 Pr[ B ∩ D ] = 5 16 � = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion A Used Car Lot Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[ B ] = 8 16 = 1 Pr[ D ] = 6 16 = 3 2 8 Pr[ B ∩ D ] = 5 16 � = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion Outline Introduction to Indepencence 1 Examples 2 Conclusion 3

Introduction to Indepencence Examples Conclusion Important Concepts Things to Remember from Section 7-5 1 Events A and B are independent if Pr[ A ∩ B ] = Pr[ A ] · Pr[ B ] 2 Events A and B are independent if Pr[ A | B ] = Pr[ A ]

Introduction to Indepencence Examples Conclusion Important Concepts Things to Remember from Section 7-5 1 Events A and B are independent if Pr[ A ∩ B ] = Pr[ A ] · Pr[ B ] 2 Events A and B are independent if Pr[ A | B ] = Pr[ A ]