Sample Spaces and Assignment of Probability
Sample Spaces and Assignment of Probability
Sample Spaces and Assignment of Probability
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<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionMATH 105: Finite Mathematics7-1: <strong>Sample</strong> <strong>Spaces</strong> <strong>and</strong> <strong>Assignment</strong> <strong>of</strong> <strong>Probability</strong>Pr<strong>of</strong>. Jonathan DuncanWalla Walla CollegeWinter Quarter, 2006
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionOutline1 <strong>Probability</strong>2 <strong>Sample</strong> <strong>Spaces</strong>3 Assigning <strong>Probability</strong>4 Conclusion
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionOutline1 <strong>Probability</strong>2 <strong>Sample</strong> <strong>Spaces</strong>3 Assigning <strong>Probability</strong>4 Conclusion
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionIntroduction to <strong>Probability</strong>Many real world events can be considered chance or r<strong>and</strong>om. Theymay be deterministic, but we can not know or comprehend all thefactors which determine the outcome.ExampleYou flip a coin. Air current, the arrangement <strong>of</strong> the coin on yourfinger, the force <strong>of</strong> your flip, <strong>and</strong> other factors all go together todetermine the outcome <strong>of</strong> Heads or Tails.For any one toss, these factors are too complicated to take intoaccount, <strong>and</strong> the outcome appears r<strong>and</strong>om. Since the outcome isheads roughly half the time, we assign the following probabilities:Pr[H] = 1 2Pr[T ] = 1 2
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionIntroduction to <strong>Probability</strong>Many real world events can be considered chance or r<strong>and</strong>om. Theymay be deterministic, but we can not know or comprehend all thefactors which determine the outcome.ExampleYou flip a coin. Air current, the arrangement <strong>of</strong> the coin on yourfinger, the force <strong>of</strong> your flip, <strong>and</strong> other factors all go together todetermine the outcome <strong>of</strong> Heads or Tails.For any one toss, these factors are too complicated to take intoaccount, <strong>and</strong> the outcome appears r<strong>and</strong>om. Since the outcome isheads roughly half the time, we assign the following probabilities:Pr[H] = 1 2Pr[T ] = 1 2
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> Vocabulary<strong>Probability</strong> TermsOutcomeA particular result <strong>of</strong> an activity or event.EventA set <strong>of</strong> outcomes which share a common characteristic.<strong>Sample</strong> SpaceThe set <strong>of</strong> all possible outcomes for an experiment. This isthe universal set for the experiment.Equally Likely EventsAll events in the sample space have the same probability.
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> Vocabulary<strong>Probability</strong> TermsOutcomeA particular result <strong>of</strong> an activity or event.EventA set <strong>of</strong> outcomes which share a common characteristic.<strong>Sample</strong> SpaceThe set <strong>of</strong> all possible outcomes for an experiment. This isthe universal set for the experiment.Equally Likely EventsAll events in the sample space have the same probability.
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<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionFinding <strong>Sample</strong> <strong>Spaces</strong>One <strong>of</strong> the first tasks in finding probability is to determine thesample space for the experiment.ExampleYou flip a fair coin. What is the sample space for this experiment?ExampleYou roll a six-sided die <strong>and</strong> note the number which appears on top.What is the sample space for this experiment?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionFinding <strong>Sample</strong> <strong>Spaces</strong>One <strong>of</strong> the first tasks in finding probability is to determine thesample space for the experiment.ExampleYou flip a fair coin. What is the sample space for this experiment?S = {H, T }ExampleYou roll a six-sided die <strong>and</strong> note the number which appears on top.What is the sample space for this experiment?S = {1, 2, 3, 4, 5, 6}
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionFinding More <strong>Sample</strong> <strong>Spaces</strong>ExampleYou flip a coin <strong>and</strong> roll a die, <strong>and</strong> note the result <strong>of</strong> each. what isthe sample space for this experiment?S = {H1, H2, . . ., H6, T 1, T 2, . . ., T 6}c(S) = 2 · 6 = 12
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionFinding More <strong>Sample</strong> <strong>Spaces</strong>ExampleYou flip a coin <strong>and</strong> roll a die, <strong>and</strong> note the result <strong>of</strong> each. what isthe sample space for this experiment?S = {H1, H2, . . ., H6, T 1, T 2, . . ., T 6}c(S) = 2 · 6 = 12
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionFinding More <strong>Sample</strong> <strong>Spaces</strong>ExampleYou flip a coin <strong>and</strong> roll a die, <strong>and</strong> note the result <strong>of</strong> each. what isthe sample space for this experiment?S = {H1, H2, . . ., H6, T 1, T 2, . . ., T 6}c(S) = 2 · 6 = 12
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDifferent <strong>Sample</strong> <strong>Spaces</strong> for the Same ExperimentExampleYou roll two dice <strong>and</strong> note both numbers. What is the samplespace for this experiment?ExampleYou roll two dice <strong>and</strong> note the sum <strong>of</strong> the two numbers. What isthe sample space for this experiment?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDifferent <strong>Sample</strong> <strong>Spaces</strong> for the Same ExperimentExampleYou roll two dice <strong>and</strong> note both numbers. What is the samplespace for this experiment?S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36ExampleYou roll two dice <strong>and</strong> note the sum <strong>of</strong> the two numbers. What isthe sample space for this experiment?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDifferent <strong>Sample</strong> <strong>Spaces</strong> for the Same ExperimentExampleYou roll two dice <strong>and</strong> note both numbers. What is the samplespace for this experiment?S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36ExampleYou roll two dice <strong>and</strong> note the sum <strong>of</strong> the two numbers. What isthe sample space for this experiment?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDifferent <strong>Sample</strong> <strong>Spaces</strong> for the Same ExperimentExampleYou roll two dice <strong>and</strong> note both numbers. What is the samplespace for this experiment?S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36ExampleYou roll two dice <strong>and</strong> note the sum <strong>of</strong> the two numbers. What isthe sample space for this experiment?S = {2, 3, . . ., 12} c(S) = 11
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleYou take a True/False quiz with three questions. If you treat thisquiz as an experiment, what is the sample space?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleYou take a True/False quiz with three questions. If you treat thisquiz as an experiment, what is the sample space?S = {TTT , TTF , . . ., FFF } c(S) = 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleYou take a True/False quiz with three questions. If you treat thisquiz as an experiment, what is the sample space?S = {TTT , TTF , . . ., FFF } c(S) = 8Now that we have some practice identifying sample spaces, it is timeto start assigning probabilities.
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionOutline1 <strong>Probability</strong>2 <strong>Sample</strong> <strong>Spaces</strong>3 Assigning <strong>Probability</strong>4 Conclusion
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?S = {TTT , TTF , TFT , TFF , FTT , FTF , FFT , FFF }Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?S = {TTT , TTF , TFT , TFF , FTT , FTF , FFT , FFF }A few rules before we actually assign probabilities.Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?S = {TTT , TTF , TFT , TFF , FTT , FTF , FFT , FFF }A few rules before we actually assign probabilities.Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?S = {TTT , TTF , TFT , TFF , FTT , FTF , FFT , FFF }A few rules before we actually assign probabilities.Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionTaking a QuizExampleHow likely are you to get all three answers in the True/False quizcorrect if you guess on each question?S = {TTT , TTF , TFT , TFF , FTT , FTF , FFT , FFF }A few rules before we actually assign probabilities.Rules for Assigning <strong>Probability</strong>For each outcome W , 0 ≤ Pr[W ] ≤ 1The sum <strong>of</strong> the probabilities <strong>of</strong> all outcomes is one.Equally Likely OutcomesPr[TTT ] = Pr[TTF ] = . . . = Pr[FFF ] = 1c(S) = 1 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> ModelWhen you find the sample space for an experiment <strong>and</strong> assignprobabilities to each element <strong>of</strong> the sample space, you areconstructing a probability model.ExampleA six sided die is weighted so that the 1 is twice as likely as anyother number <strong>and</strong> all other numbers are equally likely. Find theprobability model.S = { 1, 2, 3, 4, 5, 6, }2x x x x x xPr[1] = 2 7Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> ModelWhen you find the sample space for an experiment <strong>and</strong> assignprobabilities to each element <strong>of</strong> the sample space, you areconstructing a probability model.ExampleA six sided die is weighted so that the 1 is twice as likely as anyother number <strong>and</strong> all other numbers are equally likely. Find theprobability model.S = { 1, 2, 3, 4, 5, 6, }2x x x x x xPr[1] = 2 7Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> ModelWhen you find the sample space for an experiment <strong>and</strong> assignprobabilities to each element <strong>of</strong> the sample space, you areconstructing a probability model.ExampleA six sided die is weighted so that the 1 is twice as likely as anyother number <strong>and</strong> all other numbers are equally likely. Find theprobability model.S = { 1, 2, 3, 4, 5, 6, }2x x x x x xPr[1] = 2 7Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> Conclusion<strong>Probability</strong> ModelWhen you find the sample space for an experiment <strong>and</strong> assignprobabilities to each element <strong>of</strong> the sample space, you areconstructing a probability model.ExampleA six sided die is weighted so that the 1 is twice as likely as anyother number <strong>and</strong> all other numbers are equally likely. Find theprobability model.S = { 1, 2, 3, 4, 5, 6, }2x x x x x xPr[1] = 2 7Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionProbabilities <strong>of</strong> EventsTo find the probability <strong>of</strong> an event in sample spaces with equallylikely outcomes, we use the following probability formula.<strong>Probability</strong> <strong>of</strong> an EventIf E is a subset <strong>of</strong> a sample space S in which all outcomes areequally likely, thenPr[E] = c(E)c(S)ExampleYou guess on all 3 questions in the True/False quiz seen earlier.What is the probability that you miss one?E = {TTF , TFT , FTT }Pr[E] = c(E)c(S) = 3 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionProbabilities <strong>of</strong> EventsTo find the probability <strong>of</strong> an event in sample spaces with equallylikely outcomes, we use the following probability formula.<strong>Probability</strong> <strong>of</strong> an EventIf E is a subset <strong>of</strong> a sample space S in which all outcomes areequally likely, thenPr[E] = c(E)c(S)ExampleYou guess on all 3 questions in the True/False quiz seen earlier.What is the probability that you miss one?E = {TTF , TFT , FTT }Pr[E] = c(E)c(S) = 3 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionProbabilities <strong>of</strong> EventsTo find the probability <strong>of</strong> an event in sample spaces with equallylikely outcomes, we use the following probability formula.<strong>Probability</strong> <strong>of</strong> an EventIf E is a subset <strong>of</strong> a sample space S in which all outcomes areequally likely, thenPr[E] = c(E)c(S)ExampleYou guess on all 3 questions in the True/False quiz seen earlier.What is the probability that you miss one?E = {TTF , TFT , FTT }Pr[E] = c(E)c(S) = 3 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionProbabilities <strong>of</strong> EventsTo find the probability <strong>of</strong> an event in sample spaces with equallylikely outcomes, we use the following probability formula.<strong>Probability</strong> <strong>of</strong> an EventIf E is a subset <strong>of</strong> a sample space S in which all outcomes areequally likely, thenPr[E] = c(E)c(S)ExampleYou guess on all 3 questions in the True/False quiz seen earlier.What is the probability that you miss one?E = {TTF , TFT , FTT }Pr[E] = c(E)c(S) = 3 8
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDrawing Balls from an UrnExampleA jar contains 8 balls: 4 green, 3 blue, <strong>and</strong> 1 red. You pick oneball at r<strong>and</strong>om. Find:1 The probability the ball you draw is green.2 The probability the ball you draw is not red.ExampleAn urn contains 3 balls: one red, one green, <strong>and</strong> one yellow. Youdraw the balls out one-by-one at r<strong>and</strong>om. What is the probabilitythat the yellow ball is not drawn drawn last?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionDrawing Balls from an UrnExampleA jar contains 8 balls: 4 green, 3 blue, <strong>and</strong> 1 red. You pick oneball at r<strong>and</strong>om. Find:1 The probability the ball you draw is green.2 The probability the ball you draw is not red.ExampleAn urn contains 3 balls: one red, one green, <strong>and</strong> one yellow. Youdraw the balls out one-by-one at r<strong>and</strong>om. What is the probabilitythat the yellow ball is not drawn drawn last?
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionRolling Two DiceExampleYou roll two fair six-sided dice <strong>and</strong> note the sum <strong>of</strong> the rolls. Findeach probability.1 Pr[ sum is 7 ]2 Pr[ sum is 4 ]3 Pr[ sum is 4 or 7 ]4 Pr[ sum is 4 <strong>and</strong> 7 ]
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionOutline1 <strong>Probability</strong>2 <strong>Sample</strong> <strong>Spaces</strong>3 Assigning <strong>Probability</strong>4 Conclusion
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionImportant ConceptsThings to Remember from Section 7-11 <strong>Probability</strong> Vocabulary: Outcomes, Events, <strong>Sample</strong> <strong>Spaces</strong>2 Finding <strong>Sample</strong> <strong>Spaces</strong>3 Building <strong>Probability</strong> Models4 Assigning Probabilities to Events
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionImportant ConceptsThings to Remember from Section 7-11 <strong>Probability</strong> Vocabulary: Outcomes, Events, <strong>Sample</strong> <strong>Spaces</strong>2 Finding <strong>Sample</strong> <strong>Spaces</strong>3 Building <strong>Probability</strong> Models4 Assigning Probabilities to Events
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionImportant ConceptsThings to Remember from Section 7-11 <strong>Probability</strong> Vocabulary: Outcomes, Events, <strong>Sample</strong> <strong>Spaces</strong>2 Finding <strong>Sample</strong> <strong>Spaces</strong>3 Building <strong>Probability</strong> Models4 Assigning Probabilities to Events
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionImportant ConceptsThings to Remember from Section 7-11 <strong>Probability</strong> Vocabulary: Outcomes, Events, <strong>Sample</strong> <strong>Spaces</strong>2 Finding <strong>Sample</strong> <strong>Spaces</strong>3 Building <strong>Probability</strong> Models4 Assigning Probabilities to Events
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionImportant ConceptsThings to Remember from Section 7-11 <strong>Probability</strong> Vocabulary: Outcomes, Events, <strong>Sample</strong> <strong>Spaces</strong>2 Finding <strong>Sample</strong> <strong>Spaces</strong>3 Building <strong>Probability</strong> Models4 Assigning Probabilities to Events
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionNext Time. . .Since probabilities are based on sets: the sample space <strong>and</strong> events,it is conceivable that tools used to work with sets would also beimportant in working with probabilities.Indeed, next time we will use rules for combining sets <strong>and</strong> VennDiagrams to help solve probability problems.For next timeRead Section 7-2 (pp 376-384)Do Problem Sets 7-1 A,B
<strong>Probability</strong> <strong>Sample</strong> <strong>Spaces</strong> Assigning <strong>Probability</strong> ConclusionNext Time. . .Since probabilities are based on sets: the sample space <strong>and</strong> events,it is conceivable that tools used to work with sets would also beimportant in working with probabilities.Indeed, next time we will use rules for combining sets <strong>and</strong> VennDiagrams to help solve probability problems.For next timeRead Section 7-2 (pp 376-384)Do Problem Sets 7-1 A,B
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