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- 2. Probability---Introduction One of the most important disciplines in Computer Science (CS). Algorithm Design and Game Theory
- 3. Probability---Introduction---Cont. But it is also probably the least well understood Human intuition and Random events Goal:
- 4. Probability Contents Basic definitions and an elementary 4-step process Counting Conditional probability and the concept of
- 5. Probability Let’s Make a Deal The famous game show (you might have seen this problem in
- 6. Probability Precise Description The car is equally likely to be hidden behind the three doors. Equally
- 7. Probability Precise Description The car is equally likely to be hidden behind the three doors. The
- 8. Probability Solving Problems Involving Probability Model the situation mathematically Solve the resulting mathematical problem
- 9. Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes
- 10. Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes
- 11. Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes
- 12. Probability Solving Problems Involving Probability Step 1: Finding the sample space Set of all possible outcomes
- 13. Probability Finding the Sample Space Every possible value of these quantities is called an outcome. And
- 14. Probability Finding the Sample Space Every possible value of these quantities is called an outcome. And
- 15. Probability Possibility Tree The first quantity in our example is the door concealing the car Represent
- 16. Probability Possibility Tree --- Cont. The car can be at any of these three locations
- 17. Probability Possibility Tree --- Cont. The car can be at any of these three locations For
- 18. Probability Possibility Tree --- Cont. The car can be at any of these three locations For
- 19. Probability Possibility Tree --- Cont.
- 20. Probability Finding The Sample Space The leaves of the possibility tree represent the outcomes of a
- 22. Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest:
- 23. Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest: Remember, we want to
- 24. Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest: Remember, we want to
- 25. Probability Solving Problems Involving Probability Step 2: Defining the Events of Interest: Remember, we want to
- 28. Probability Solving Problems Involving Probability Coming back to our example We want to know: “What is
- 29. Probability Solving Problems Involving Probability---Cont. Notice: Half of the outcomes are checked. Does this mean that
- 30. Probability Solving Problems Involving Probability---Cont. Step 3: Determining Outcome Probability Assign Edge Probabilities Compute Outcome Probabilities
- 31. Probability Equally likely probability formula E: the equally likely event S: the sample space
- 32. Probability Solving Problems Involving Probability---Cont. Step 3: Determining Outcome Probability Assign Edge Probabilities Compute Outcome Probabilities
- 33. Probability Edge Probabilities To understand, let’s analyze the path leading to the leaf node (A, A,
- 34. Probability Multiplication Rule The probability that Events A and B both occur is equal to the
- 35. Probability Outcome Probabilities To understand, let’s analyze the probability of the outcome (A, A, B).
- 37. Probability Summary To solve problems involving probability, that is, “what is the probability that … ?”
- 38. Probability Uniform Sample Space Strange Dice If we picked dices (a) and (b), rolled them once,
- 39. Probability Applying Four-Step Method When the probability of every outcome is the same, we say such
- 41. Probability Applying Four-Step Method Example--- Cont. What about the following: (a) vs. (c) (b) vs. (c)
- 43. Probability Counting Rules of counting the elements in a set
- 44. Probability The Addition Rule The basic rule underlying the calculation of the number of elements in
- 45. Probability The Addition Rule---Cont. Example: A computer access password consists of from one to three letters
- 48. Probability The Difference Rule An important consequence of the addition rule is the fact that if
- 49. Probability The Difference Rule---Cont. The difference rule is illustrated below.
- 51. Probability The Difference Rule---Cont. Example: A typical PIN (personal identification number) is a sequence of any
- 52. Probability The Difference Rule---Cont. a. How many PINs contain repeated symbols? Let’s use the board to
- 53. Probability The Difference Rule---Cont. Example --- Cont.: There are 364 = 1,679,616 PINs when repetition is
- 54. Probability The Difference Rule---Cont. Example --- Cont.: There are 364 = 1,679,616 PINs when repetition is
- 55. Probability The Difference Rule---Cont. b. If all PINs are equally likely, what is the probability that
- 57. Probability The Difference Rule---Cont. An alternative solution to Example 3(b) is based on the observation that
- 64. Probability The Difference Rule---Cont. This solution illustrates a more general property of probabilities: that the probability
- 65. Probability The Inclusion/Exclusion Rule The addition rule says how many elements are in a union of
- 70. Further Counting Counting Subsets of a Set: Combinations: Look at these examples: In how many ways,
- 71. Further Counting Counting Subsets of a Set: Combinations: Look at these examples: In how many ways,
- 73. Why Count Subsets of Set? Example: Suppose we select 5 cards at random from a deck
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