Lecture 4-Probability Slides

12 example rolling two dice what is the probability

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Unformatted text preview: ent “the sum is less than 5”? 12 Example: Rolling two dice What is the probability of the event “the sum is less than 5”? Die 2 1 Die 1 1 2 3 4 5 6 P(“sum less than 5”) = 6/36 = 1/6 2 3 4 5 6 13 Views of Probability frequentist view: relative frequency NA n−>∞ N P (A) = lim N number of experiments NA number of experiments where A happened Bayesian view: degree of belief “What is the probability that there is life on Mars” 14 Bayesian probability From Wikipedia, the free encyclopedia. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements. This is in contrast to frequentism, which rejects degree-of-belief interpretations of mathematical probability, and assigns probabilities only to random events according to their relative frequencies of occurrence. The Bayesian interpretation of probability allows probabilities assigned to random events, but also allows the assignment of probabilities to any other kind of statement. Whereas a frequentist and a Bayesian might both assign probability 1/2 to the event of getting a head when a coin is tossed, a Bayesian might assign probability 1/2 to personal belief in the proposition that there was life on Mars a billion years ago, without intending to assert anything about any relative frequency. 15 Conditional Probability How does one event occuring affect the chance of occurence of another event occuring. “What is the probability of voting for Romney, given that you have an income of over 1 million dollars?” Probability of A given B P (A|B ) = P (A, B ) P (B ) 16 Jochen’s dice example What is P(D1=1,D2=2)? 17 Jochen’s dice example Die 2 1 2 3 4 5 6 Die 1 1 2 3 4 5 6 What is P(D1=1,D2=2)? 1/36 18 Jochen’s dice example What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 19 Jochen’s dice example Die 2 1 2 3 4 5 6 Die 1 1 2 3 4 5 6 What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 1/6 20 Jochen’s dice example What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 1/6 What is P(D1=1 | sum is even)? 21 Jochen’s dice example Die 2 1 2 3 4 5 6 Die 1 1 2 3 4 5 6 What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 1/6 What is P(D1=1 | sum is even)? 3/18=1/6 (sum is even along alternating diagonals) 22 Jochen’s dice example What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 1/6 What is P(D1=1 | sum is even)? 3/18=1/6 sum is even along alternating diagonals What is P(D1=1 | sum is 4)? 23 Jochen’s dice example Die 2 1 2 3 4 5 6 Die 1 1 2 3 4 5 6 What is P(D1=1,D2=2)? 1/36 What is P(D1=1 | D2=2)? 1/6 What is P(D1=1 | sum is even)? 3/18=1/6 sum is even along alternating diagonals What...
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This note was uploaded on 02/09/2014 for the course COGS 109 taught by Professor Staff during the Fall '08 term at UCSD.

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