Introduction to Probability (Dr. Monticino). Assignment Sheet  Read Chapters 13 and 14  Assignment #8 (Due Wednesday March 23 rd )  Chapter 13  Exercise.

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Presentation transcript:

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  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

Overview  Framework  Equally likely outcomes  Some rules

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)

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

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

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

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)

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)

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

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)