MATH 105: Finite Mathematics 7-4: Conditional Probability
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleMATH 105: Finite Mathematics7-4: Conditional ProbabilityProf. Jonathan DuncanWalla Walla CollegeWinter Quarter, 2006Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleOutline1Introduction to Conditional Probability2Some Examples3A “New” Multiplication Rule4ConclusionConclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleOutline1Introduction to Conditional Probability2Some Examples3A “New” Multiplication Rule4ConclusionConclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionExtra Information and ProbabilityIn 1991 the following problem caused quite a stir in the world ofmathematics.Monty Hall ProblemMonty Hall, the host of “Let’s Make a Deal” invites you to play agame. He presents you with three doors and tells you that two ofthe doors hide goats, and one hides a new car. You get to chooseone door and keep whatever is behind that door.You choose a door, and Monte opens one of the other two doorsto reveal a goat. He then asks you if you wish to keep your originaldoor, or switch to the other door?Play the Game
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.123Door AgoatgoatcarDoor BgoatcargoatDoor CcargoatgoatExample:You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.123Door AgoatgoatcarDoor BgoatcargoatDoor CcargoatgoatExample:You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.123Door AgoatgoatcarDoor BgoatcargoatDoor CcargoatgoatExample:You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.123Door AgoatgoatcarDoor BgoatcargoatDoor CcargoatgoatExample:You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleExtra Information and Probability, continued. . .Monty Hall SolutionYou should switch doors.123Door AgoatgoatcarDoor BgoatcargoatDoor CcargoatgoatExample:You choose Door A and have a13probability of winning.Monty eliminates a goat behind one of the other doors.Switching wins in cases 1 and 2 and looses in case 3.Thus, switching raises your probability of winning to 32 .Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement.1What is the probability that both are red?2What is the probability that both are red given that the firstis white?3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement.1What is the probability that both are red?2What is the probability that both are red given that the firstis white?3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement.1What is the probability that both are red?2What is the probability that both are red given that the firstis white?3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement.1What is the probability that both are red?C (8, 2)28 C (10, 2)452What is the probability that both are red given that the firstis white?3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement. 281 What is the probability that both are red?452What is the probability that both are red given that the firstis white?3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement. 281 What is the probability that both are red?452What is the probability that both are red given that the firstis white?This can’t happen3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement. 281 What is the probability that both are red?452What is the probability that both are red given that the firstis white? (0)3What is the probability that both are red given that the firstis red?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement. 281 What is the probability that both are red?452What is the probability that both are red given that the firstis white? (0)3What is the probability that both are red given that the firstis red?79
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityHere is another example of Conditional Probability.ExampleAn urn contains 10 balls: 8 red and 2 white. Two balls are drawnat random without replacement. 281 What is the probability that both are red?452What is the probability that both are red given that the firstis white? (0)3What is the probability that both are red given that the firstis red?In the last two questions, extra information changed the probability.
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityInformation given about one event can effect the probability of asecond event. Knowing that the first ball was white in the problemabove changed the probability that both balls were red.Conditional ProbabiltyIf A and B are events in a sample space then the probability of Ahappening given that B happens is denotedPr[A B]which is read “The probabilty of A given B”.
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionConditional ProbabilityInformation given about one event can effect the probability of asecond event. Knowing that the first ball was white in the problemabove changed the probability that both balls were red.Conditional ProbabiltyIf A and B are events in a sample space then the probability of Ahappening given that B happens is denotedPr[A B]which is read “The probabilty of A given B”.
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionVenn Diagrams and Conditional ProbabilityTo help us develop a formula for Pr[A B] we will use VennDiagrams in the following example.ExampleThe probability of an event A is 0.50. The probability of an eventB is 0.70. The probability of A B is 0.30. Find Pr[A B] andPr[B A].Pr[A B] 0.30 0.430.40 0.30Pr[B A] 0.30 0.600.30 0.20
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionVenn Diagrams and Conditional ProbabilityTo help us develop a formula for Pr[A B] we will use VennDiagrams in the following example.ExampleThe probability of an event A is 0.50. The probability of an eventB is 0.70. The probability of A B is 0.30. Find Pr[A B] andPr[B A].Pr[A B] 0.30 0.430.40 0.30Pr[B A] 0.30 0.600.30 0.20
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionVenn Diagrams and Conditional ProbabilityTo help us develop a formula for Pr[A B] we will use VennDiagrams in the following example.ExampleThe probability of an event A is 0.50. The probability of an eventB is 0.70. The probability of A B is 0.30. Find Pr[A B] andPr[B A].AC0.20.3Pr[A B] 0.30 0.430.40 0.30Pr[B A] 0.30 0.600.30 0.200.40.1
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionVenn Diagrams and Conditional ProbabilityTo help us develop a formula for Pr[A B] we will use VennDiagrams in the following example.ExampleThe probability of an event A is 0.50. The probability of an eventB is 0.70. The probability of A B is 0.30. Find Pr[A B] andPr[B A].AC0.20.3Pr[A B] 0.30 0.430.40 0.30Pr[B A] 0.30 0.600.30 0.200.40.1
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionVenn Diagrams and Conditional ProbabilityTo help us develop a formula for Pr[A B] we will use VennDiagrams in the following example.ExampleThe probability of an event A is 0.50. The probability of an eventB is 0.70. The probability of A B is 0.30. Find Pr[A B] andPr[B A].AC0.20.3Pr[A B] 0.30 0.430.40 0.30Pr[B A] 0.30 0.600.30 0.200.40.1
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConditional Probability FormulaConditional Probability FormulaLet A and B be events in a sample space. Then,Pr[A B] Pr[A B]Pr[B]Note:Pr[A B] Pr[B]c(A B)c(S)c(B)c(S) c(A B)c(B)Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConditional Probability FormulaConditional Probability FormulaLet A and B be events in a sample space. Then,Pr[A B] Pr[A B]Pr[B]Note:Pr[A B] Pr[B]c(A B)c(S)c(B)c(S) c(A B)c(B)Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleOutline1Introduction to Conditional Probability2Some Examples3A “New” Multiplication Rule4ConclusionConclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionRolling Two DiceExampleYou roll two dice and note their sum.1What is the probability of at least one 3 given that the sum issix?2What is the probability that the sum is six given that there isat least one 3?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionRolling Two DiceExampleYou roll two dice and note their sum.1What is the probability of at least one 3 given that the sum issix?2What is the probability that the sum is six given that there isat least one 3?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionRolling Two DiceExampleYou roll two dice and note their sum.1What is the probability of at least one 3 given that the sum issix?c(A B)1Pr[B A] c(A)52What is the probability that the sum is six given that there isat least one 3?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionRolling Two DiceExampleYou roll two dice and note their sum.1What is the probability of at least one 3 given that the sum issix?c(A B)1Pr[B A] c(A)52What is the probability that the sum is six given that there isat least one 3?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionRolling Two DiceExampleYou roll two dice and note their sum.1What is the probability of at least one 3 given that the sum issix?c(A B)1Pr[B A] c(A)52What is the probability that the sum is six given that there isat least one 3?Pr[A B] c(A B)1 c(B)11
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionOn a Used Car LotExampleThere are 4 vans, 2 SUVs, 6 compacts, and 3 motorcycles on aused car lot. One is chosen at random to be the “special sale”vehicle.1What is the probability the van is chosen given that the SUVsare not chosen?2What is the probability that the compact is chosen given thatonly vans or compacts are elligible?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionOn a Used Car LotExampleThere are 4 vans, 2 SUVs, 6 compacts, and 3 motorcycles on aused car lot. One is chosen at random to be the “special sale”vehicle.1What is the probability the van is chosen given that the SUVsare not chosen?2What is the probability that the compact is chosen given thatonly vans or compacts are elligible?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionOn a Used Car LotExampleThere are 4 vans, 2 SUVs, 6 compacts, and 3 motorcycles on aused car lot. One is chosen at random to be the “special sale”vehicle.1What is the probability the van is chosen given that the SUVsare not chosen?Pr[ van not SUV ] 2413What is the probability that the compact is chosen given thatonly vans or compacts are elligible?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionOn a Used Car LotExampleThere are 4 vans, 2 SUVs, 6 compacts, and 3 motorcycles on aused car lot. One is chosen at random to be the “special sale”vehicle.1What is the probability the van is chosen given that the SUVsare not chosen?Pr[ van not SUV ] 2413What is the probability that the compact is chosen given thatonly vans or compacts are elligible?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionOn a Used Car LotExampleThere are 4 vans, 2 SUVs, 6 compacts, and 3 motorcycles on aused car lot. One is chosen at random to be the “special sale”vehicle.1What is the probability the van is chosen given that the SUVsare not chosen?Pr[ van not SUV ] 2413What is the probability that the compact is chosen given thatonly vans or compacts are elligible?Pr[ compact compact or van ] 610
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleOutline1Introduction to Conditional Probability2Some Examples3A “New” Multiplication Rule4ConclusionConclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleRevising the FormulaRevised Counditional Probability FormulaWe have seen the formula for conditional probability:Pr[A B] Pr[A B]Pr[B]Multiplying both sides by Pr[B] yields:Pr[A B] Pr[B] · Pr[A B]Note:The second formula above allows us to use tree diagrams tocompute probabilities using tree diagrams.Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleRevising the FormulaRevised Counditional Probability FormulaWe have seen the formula for conditional probability:Pr[A B] Pr[A B]Pr[B]Multiplying both sides by Pr[B] yields:Pr[A B] Pr[B] · Pr[A B]Note:The second formula above allows us to use tree diagrams tocompute probabilities using tree diagrams.Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleProbability on a Tree DiagramExampleTwo urns contain colored balls. The first has 2 white and 3 redballs, and the second has 1 white, 2 red, and 3 yellow balls. Oneurn is selected at random and then a ball is drawn. Construct atree diagram showing all probabilities for this experiment.Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionMore Probabilities on Tree DiagramsExampleAn experiment consists of 3 steps. First, an unfair coin withPr[H] 31 is flipped. If a heads appears, a ball is drawn from urn#1 which contains 2 white and 3 red balls. If a tails is flipped, aball is drawn from urn #2 which contains 4 white and 2 red balls.Finally, a ball is drawn from the other urn. Construct a treediagram to help answer the following questions.1What is Pr[HW W ]?2What is Pr[ both balls red H flipped ]?3What is Pr[ 1st ball red T flipped ]?4What is Pr[ last ball red ]?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionMore Probabilities on Tree DiagramsExampleAn experiment consists of 3 steps. First, an unfair coin withPr[H] 31 is flipped. If a heads appears, a ball is drawn from urn#1 which contains 2 white and 3 red balls. If a tails is flipped, aball is drawn from urn #2 which contains 4 white and 2 red balls.Finally, a ball is drawn from the other urn. Construct a treediagram to help answer the following questions.1What is Pr[HW W ]?2What is Pr[ both balls red H flipped ]?3What is Pr[ 1st ball red T flipped ]?4What is Pr[ last ball red ]?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionMore Probabilities on Tree DiagramsExampleAn experiment consists of 3 steps. First, an unfair coin withPr[H] 31 is flipped. If a heads appears, a ball is drawn from urn#1 which contains 2 white and 3 red balls. If a tails is flipped, aball is drawn from urn #2 which contains 4 white and 2 red balls.Finally, a ball is drawn from the other urn. Construct a treediagram to help answer the following questions.1What is Pr[HW W ]?2What is Pr[ both balls red H flipped ]?3What is Pr[ 1st ball red T flipped ]?4What is Pr[ last ball red ]?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionMore Probabilities on Tree DiagramsExampleAn experiment consists of 3 steps. First, an unfair coin withPr[H] 31 is flipped. If a heads appears, a ball is drawn from urn#1 which contains 2 white and 3 red balls. If a tails is flipped, aball is drawn from urn #2 which contains 4 white and 2 red balls.Finally, a ball is drawn from the other urn. Construct a treediagram to help answer the following questions.1What is Pr[HW W ]?2What is Pr[ both balls red H flipped ]?3What is Pr[ 1st ball red T flipped ]?4What is Pr[ last ball red ]?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionMore Probabilities on Tree DiagramsExampleAn experiment consists of 3 steps. First, an unfair coin withPr[H] 31 is flipped. If a heads appears, a ball is drawn from urn#1 which contains 2 white and 3 red balls. If a tails is flipped, aball is drawn from urn #2 which contains 4 white and 2 red balls.Finally, a ball is drawn from the other urn. Construct a treediagram to help answer the following questions.1What is Pr[HW W ]?2What is Pr[ both balls red H flipped ]?3What is Pr[ 1st ball red T flipped ]?4What is Pr[ last ball red ]?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionPreparing for Next TimeThe next two sections will study questions such as those below inmore detail.ExampleIn the previous example, find Pr[ last ball red H flipped ] andPr[ last ball red T flipped ]. Does the result of the coin tosschange the probability that the last ball is red?ExampleAgain using the previous example, findPr[ H flipped last ball red ]. Can the tree diagram be used to findthis probability?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionPreparing for Next TimeThe next two sections will study questions such as those below inmore detail.ExampleIn the previous example, find Pr[ last ball red H flipped ] andPr[ last ball red T flipped ]. Does the result of the coin tosschange the probability that the last ball is red?ExampleAgain using the previous example, findPr[ H flipped last ball red ]. Can the tree diagram be used to findthis probability?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleConclusionPreparing for Next TimeThe next two sections will study questions such as those below inmore detail.ExampleIn the previous example, find Pr[ last ball red H flipped ] andPr[ last ball red T flipped ]. Does the result of the coin tosschange the probability that the last ball is red?ExampleAgain using the previous example, findPr[ H flipped last ball red ]. Can the tree diagram be used to findthis probability?
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleOutline1Introduction to Conditional Probability2Some Examples3A “New” Multiplication Rule4ConclusionConclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleImportant ConceptsThings to Remember from Section 7-41Conditional Probability Formula:Pr[A B] 2Pr[A B]Pr[B]Using tree diagrams for probability:Pr[A B] Pr[B] · Pr[A B]Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleImportant ConceptsThings to Remember from Section 7-41Conditional Probability Formula:Pr[A B] 2Pr[A B]Pr[B]Using tree diagrams for probability:Pr[A B] Pr[B] · Pr[A B]Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleImportant ConceptsThings to Remember from Section 7-41Conditional Probability Formula:Pr[A B] 2Pr[A B]Pr[B]Using tree diagrams for probability:Pr[A B] Pr[B] · Pr[A B]Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleNext Time. . .Next time we will introduce the concept of “independent events”and how they relate to conditional probabilities.For next timeRead Section 7-5Prepare for Quiz on 7-4Conclusion
Introduction to Conditional ProbabilitySome ExamplesA “New” Multiplication RuleNext Time. . .Next time we will introduce the concept of “independent events”and how they relate to conditional probabilities.For next timeRead Section 7-5Prepare for Quiz on 7-4Conclusion
Introduction to Conditional Probability Some Examples A “New” Multiplication Rule Conclusion Conditional Probability Here is another example of Conditional Probability. Example An urn contains 10 balls: 8 red and 2 white. Two balls are drawn at random without replacement. 1 What is the proba
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