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Understanding Probability Models: Rules, Principles, and Applications

Dive into the world of probability models with this comprehensive guide, covering sample spaces, event probabilities, rules, and calculations. Explore examples like rolling dice, tree diagrams, Benford's Law, and more to enhance your understanding.

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Understanding Probability Models: Rules, Principles, and Applications

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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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