MATH 105: Finite Mathematics 7-1: Sample Spaces and Assignment of Probability

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1 MATH 105: Finite Mathematics 7-1: Sample Spaces and Assignment of Probability Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

2 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

3 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

4 Introduction to Probability Many real world events can be considered chance or random. They may be deterministic, but we can not know or comprehend all the factors which determine the outcome. You flip a coin. Air current, the arrangement of the coin on your finger, the force of your flip, and other factors all go together to determine the outcome of Heads or Tails. For any one toss, these factors are too complicated to take into account, and the outcome appears random. Since the outcome is heads roughly half the time, we assign the following probabilities: Pr[H] = 1 2 Pr[T ] = 1 2

5 Introduction to Probability Many real world events can be considered chance or random. They may be deterministic, but we can not know or comprehend all the factors which determine the outcome. You flip a coin. Air current, the arrangement of the coin on your finger, the force of your flip, and other factors all go together to determine the outcome of Heads or Tails. For any one toss, these factors are too complicated to take into account, and the outcome appears random. Since the outcome is heads roughly half the time, we assign the following probabilities: Pr[H] = 1 2 Pr[T ] = 1 2

6 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

7 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

8 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

9 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

10 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

11 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

12 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

13 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

14 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

15 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

16 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6}

17 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

18 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

19 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

20 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

21 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

22 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

23 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment? S = {2, 3,..., 12} c(s) = 11

24 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space?

25 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space? S = {TTT, TTF,..., FFF } c(s) = 8

26 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space? S = {TTT, TTF,..., FFF } c(s) = 8 Now that we have some practice identifying sample spaces, it is time to start assigning probabilities.

27 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

28 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

29 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

30 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct 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 Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

31 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct 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 Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

32 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct 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 Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

33 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct 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 Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

34 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

35 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

36 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

37 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

38 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You 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

39 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You 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

40 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You 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

41 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You 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

42 Drawing Balls from an Urn A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick one ball at random. Find: 1 The probability the ball you draw is green. 2 The probability the ball you draw is not red. An urn contains 3 balls: one red, one green, and one yellow. You draw the balls out one-by-one at random. What is the probability that the yellow ball is not drawn drawn last?

43 Drawing Balls from an Urn A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick one ball at random. Find: 1 The probability the ball you draw is green. 2 The probability the ball you draw is not red. An urn contains 3 balls: one red, one green, and one yellow. You draw the balls out one-by-one at random. What is the probability that the yellow ball is not drawn drawn last?

44 Rolling Two Dice You roll two fair six-sided dice and note the sum of the rolls. Find each probability. 1 Pr[ sum is 7 ] 2 Pr[ sum is 4 ] 3 Pr[ sum is 4 or 7 ] 4 Pr[ sum is 4 and 7 ]

45 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

46 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

47 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

48 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

49 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

50 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

51 Next Time... Since probabilities are based on sets: the sample space and events, it is conceivable that tools used to work with sets would also be important in working with probabilities. Indeed, next time we will use rules for combining sets and Venn Diagrams to help solve probability problems. For next time Read Section 7-2 (pp ) Do Problem Sets 7-1 A,B

52 Next Time... Since probabilities are based on sets: the sample space and events, it is conceivable that tools used to work with sets would also be important in working with probabilities. Indeed, next time we will use rules for combining sets and Venn Diagrams to help solve probability problems. For next time Read Section 7-2 (pp ) Do Problem Sets 7-1 A,B

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