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# Notes1 - Math 21a Biochem PROBABILITY Fall 2004 1...

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Math 21a Biochem PROBABILITY Fall 2004 1 Introduction Text: Probability by Jim Pitman, New York: Springer-Verlag c1993. Found on reserve in the Cabot Library. This is a brief outline of the last part of the course. It is intended as a list of topics and examples rather than a complete set of lecture notes. It should provide a framework that you can fill in with your lecture notes, problem sets, and if needed the text on reserve in Cabot library. The outline is broken up into the topics we have covered. Note that different topics take different amount of lecture time to cover; some less than, some more than one class. Please don’t hesitate to contact us if you have questions about the material in the course. 2 Course outline: Introduction through Conditional Proba- bility 2.1 Motivation and Introduction to notation The virus game Equally likely events Coin tossing { HH, HT, TH, TT } Throwing fair dice Winning games of chance: example the lottery or rock/paper/scissors Law of averages Relative frequency of an event: the ratio measuring how often something occurs in a sequence of observations. (Example number of heads in n coin tosses.) General rule: relative frequencies based on larger numbers of observations fluc- tuate less than those of small numbers of observations. 1

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This is the empirical law of averages: the relative frequency of an event based on n trials stabilizes as n gets larger and larger (assuming constant conditions). Other events Gender of children B/G. (Total probability one.) A family has two children. You know that one of the children is a boy. What is the probability the other child is a boy? { BB, BG, GB, GG } Lethal Doses (LD50). This is the amount of poison you need to consume to have a 50% chance of dying. (Measured as a percentage of body weight.) Dartboards: you are equally likely to hit each point on a dart board. So, P(hitting region)=Area(region)/Area(dartboard). (See figure.) The language of events in probability Experiment Outcome, outcome space Event Dictionary between the language of events and sets See figure from Examples above used to illustrate the following Outcome space Ω Event, impossible event Not A Either A or B Both A and B A and B are mutually exclusive The probability set function The probability set function is a function that assigns to an event, the real number that reflects the likelihood of that event: denoted P( A ). Properties: * It is always non-negative * The outcome space has probability one: P(Ω) = 1 * For any event A , 0 P( A ) 1 2
Example: What is the probability you will end up in a particular Harvard House? This is a randomized process. (Think of Quincy vs Dunster house - how many of the available slots are there in each?) Definition of partitions: Event B is partitioned into n events B 1 , . . . , B n if B = B 1 . . . B n and the events B 1 , . . . , B n are mutually exclusive (that is every outcome is in one and only one of the B i ).

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