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Probability Models

Probability Models. Section 6.2. Probability Models. The sample space S of a random phenomenon is the set of all possible outcomes. The event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

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Probability Models

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  1. Probability Models Section 6.2

  2. Probability Models • The sample space S of a random phenomenon is the set of all possible outcomes. • The event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. • A probability model is a mathematical description of a random phenomenon consisting of two: a sample space S and a way of assigning probabilities to events.

  3. Examples

  4. Sample space for rolling two-dice • Make an outcome diagram for rolling two dice

  5. The 36 possible outcomes in rolling two dice.

  6. Sample Space for rolling two dice • S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

  7. Lets Practice: • Provide a sample space for random digits from table B. • Provide a sample space for flipping a coin and rolling a die.

  8. Random Digit • S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

  9. Coin and Die • S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} • Now make a tree diagram for the outcomes

  10. Multiplication Principle • If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a x b number of ways.

  11. Homework • Problems 11, 12, 17, 18

  12. Probability Rules • Rule 1: the probability P(A) of any event A is between 0 and 1 inclusive. • Rule 2: If S is the sample space in a probability model, then P(S) = 1 • Rule 3: the Complement of any event A is the event that A does not occur. Complement rule • P(Ac) = 1 – P(A)

  13. More rules • Rule 4: Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common. • Addition rule for disjoint events • P(A or B) = P(A) + P(B)

  14. Independent Events • Rule 5: P(A and B) = P(A)P(B) • Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.

  15. Example • Marital status:

  16. What is the probability of a woman not being married? • P(not married) = 1 – P(married) • = 1 – 0.622 = 0.378

  17. Which two events are disjoint? • Never married and divorced • P(never married or divorced) = P(never married) + P(divorced) • = 0.298 + 0.075 • = 0.373

  18. Benford’s Law • The first digits of numbers in legitimate records often follow a distribution known as Benford’s Law. These records are tax returns, payment records, invoices, expense account claims, etc.

  19. Benford’s Law

  20. Consider the events: • A = {first digit is 1} • B = {first digit is 6 or greater}

  21. Find probabilities • First digit = P(A) = 0.301 • P(B) = P(6) + P(7) + P(8) + P(9) • = 0.067+0.058+0.051+0.046 • = 0.222

  22. What about the probability that a digit is anything other than a 1? • P(Ac) = 1 – P(A) = 0.699

  23. Disjoint events • What is the probability that the first digit is 1 or is 6 or greater? • P (A or B) = P(A) + P(B) = 0.523

  24. Random digits • What is the probability that a randomly chosen first digit is 6 or greater? • P(B) = 1/9 + 1/9 + 1/9 + 1/9 • = 0.444

  25. Assignment • Problems 19, 22, 26, 28, 32, 34, 36, 39, 41 • Due next class meeting.

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