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Introduction to Probability (Dr. Monticino)

Introduction to Probability (Dr. Monticino). Assignment Sheet. Read Chapters 13 and 14 Assignment #8 (Due Wednesday March 23 rd ) Chapter 13 Exercise Set A: 1-5; Exercise Set B: 1-3 Exercise Set C: 1-4,7; Exercise Set D: 1,3,4 Review Exercises: 2,3, 4,5,7,8,9,11 Chapter 14

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Introduction to Probability (Dr. Monticino)

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  1. Introduction to Probability(Dr. Monticino)

  2. Assignment Sheet • Read Chapters 13 and 14 • Assignment #8 (Due Wednesday March 23rd ) • Chapter 13 • Exercise Set A: 1-5; Exercise Set B: 1-3 • Exercise Set C: 1-4,7; Exercise Set D: 1,3,4 • Review Exercises: 2,3, 4,5,7,8,9,11 • Chapter 14 • Exercise Set A: 1-4; Exercise Set B: 1-4, 5 • Exercise Set C: 1,3,4,5; Exercise Set D: 1 (just calculate probabilities) • Review Exercises: 3,4,7,8,9,12

  3. Overview • Framework • Equally likely outcomes • Some rules

  4. Probability Framework • The sample space, , is the set of all outcomes from an experiment • A probability measure assigns a number to each subset (event) of the sample space, such that • 0  P(A)  1 • P( ) = 1 • If A and B are mutually exclusive (disjoint) subsets, then P(A  B) = P(A) + P(B) (addition rule)

  5. Equally Likely Outcomes • Outcomes from an experiment are said to be equally likely if they all have the same probability. • If there are n outcomes in the experiment then the outcomes being equally likely means that each outcome has probability 1/n • If there are k outcomes in an event, then the event has probability k/n • “Fair” is often used synonymously for equally likely

  6. Examples • Roll a fair die • Probability of a 5 coming up • Probability of an even number coming up • Probability of an even number or a 5 • Roll two fair die • Probability both come up “1” (double ace) • Probability of a sum of 7 • Probability of a sum of 7 or 11

  7. More Examples • Spin a roulette wheel once • Probability of “11” • Probability of “red”; probability of “black”; probability of not winning if bet on “red” • Draw one card from a well-shuffled deck of cards • Probability of drawing a king • Probability of drawing heart • Probability of drawing king of hearts

  8. Conditional Probability • All probabilities are conditional • They are conditioned based on the information available about the experiment • Conditional probability provides a formal way for conditioning probabilities based on new information • P(A | B) = P(A  B)/P(B) • P(A B) = P(A | B)  P(B)

  9. Multiplication Rule • The probability of the intersection of two events equals the probability of the first multiplied by the probability of the second given that the first event has happened • P(A B) = P(A | B)  P(B)

  10. Examples • Suppose an urn contains 5 red marbles and 8 green marbles • Probability of red on first draw • Red on second, given red on first (no replacement) • Red on first and second

  11. Independence • Intuitively, two events are independent if information that one occurred does not affect the probability that the other occurred • More formally, A and B are independent if • P(A | B) = P(A) • P(B | A) = P(B) • P(AB) = P(A)P(B) (Dr. Monticino)

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