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Introduction to Probability (Dr. Monticino) 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 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(AB) = P(A)P(B) (Dr. Monticino)

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