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AMS 311, Lecture 7 February15, 2001 Administrative Announcements: 1. Two problem quiz next Thursday, one combinatorial problem and one Bayes’ Theorem problem. 2. Our TA is Mr. Taewon Lee. His office hours are W and F from 3 to 4 pm in Math Tower: 3-129A. His e-mail address is [email protected]. 3. Chapter Four homework (due Feb 22): Starting on page 140: 2, 9*, 10, 14; starting on page 148: 4, 7*, 12; starting on page 159: 2, 6, 8*, 16; starting on page 168: 4, 8. Chapter Two Problems Section 2.2: 30. What is the probability that a random r digit number ( r 3 ) contains at least one 0, at least one 1, and at least one 2? Example problem (also related to Fisher’s Tea-tasting Lady) The Great Carsoni, a magician, claims to have extrasensory perception. In order to test this claim, he is asked to identify the 4 red cards our of 4 red and 4 black cards which are laid face down on the table. The Great Carsoni correctly identifies 3 of the red cards and incorrectly identifies 1 of the black cards. Therefore, he claims to have proved his point. What is the probability that the Great Carsoni would have correctly identified at least 3 of the red cards if he were, in fact, guessing? (Regard the 4 cards selected by the Great Carsoni as an unordered sample of size 4). Review of last class: The following is always true: Theorem 3.2. (Generalization of the Law of Multiplication): If P( A1 A2 An1 ) 0, then P( A1 A2 A3 An1 An ) P( A1 ) P( A2 | A1 ) P( A3 | A2 A1 ) P( An | A1 A2 An1 ). Definition: Let {B1 , B2 ,, Bn } be a set of nonempty subsets of the sample space S of an experiment. If the events B1 , B2 ,, Bn are mutually exclusive and n i 1 Bi S , the set {B1 , B2 ,, Bn } is called a partition of S. Theorem 3.4. Generalized Law of Total Probabililty If {B1 , B2 ,, Bn } is a partition of the sample space of an experiment and P(Bi)>0 for i 1,2, , n , then for any event A of S, n P( A) P ( A | Bi ) P( Bi ). i 1 Bayes’ Theorem Let {B1 , B2 , , Bn } be a partition of the sample space S of an experiment. If for i 1, 2, , n, P( Bi ) 0, then for any event A of S with P(A)>0, P( A| Bk ) P( Bk ) P( Bk | A) . P( A| B1 ) P( B1 ) P( A| B2 ) P( B2 ) P( A| Bn ) P( Bn ) Example problem: Diseases D1, D2, and D3 cause symptom A with probabilities 0.5, 0.7, and 0.8, respectively. If 5% of a population have disease D1, 2% of a population have disease D2, and 3.5% of a population have disease D3, what percent of the population have symptom A? Assume that the only possible causes of symptom A are D1, D2, and D3 and that no one carries more than one of these three diseases. Definition: Two events A and B are called independent if P( AB) P( A) P( B). If two events are not independent, they are called dependent. If A and B are independent, we say that {A, B} is an independent set of events. Reasons for events to be independent: Device has been constructed to have independent outcomes (roulette wheels, etc.). A sample has been taken following the precise rules. Experimental units have been randomly assigned to treatments. To show two events are independent, apply the definition. Chapter Four: Distribution Functions and Discrete Random Variables Definition: Let S be the sample space of an experiment. A real-valued function X : S R is called a random variable of the experiment if, for each interval I R, {s: X ( s) I } is an event. Definition: If X is a random variable, then the function F defined on ( , ) by F (t ) P( X t ) is called the distribution function of X. I use the term cdf (cumulative distribution function) rather than distribution function.