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Chapters 14-15 Elements of probability theory • (probabilistic) trial situation in which one of a collection of possible outcomes could occur, but precisely which one cannot be predicted with certainty • probability long-run relative frequency of an event; probabilities are estimated in three principal ways: ◦ empirically – by calculating the fraction of attempted trials in which a desired outcome has occurred ◦ theoretically – by using a mathematical model to describe the likelihood of occurrence ◦ subjectively – by making an “educated guess” • Law of Large Numbers The relative frequency of an event over a very long string of trials approaches its true probability as the number of trials grows ever larger. 1 Chapters 14-15 • disjoint events events which have no outcomes in common, that is, can never occur simultaneously • independent events events one of whose outcomes has no influence on the outcomes of the other 2 Chapters 14-15 Formal rules of probability 1. Notation: The probability of event A is labeled P (A). 2. Probability measures likelihood : P (A) lies between 0 and 1 for any event A. 3. Something has to happen: Where S is the event consisting of the set of all possible outcomes, P (S) = 1. 4. Equally likely outcomes have equal probabilities: If there are n equally likely possible outcomes and event A includes exactly k of these outcomes, then P (A) = k/n. 5. Complementary events have complementary probabilities: P (AC ) = 1 − P (A). 6. Addition rule for disjoint events: If A and B are disjoint events, then P (A or B) = P (A) + P (B). 7. Multiplication rule for independent events: If A and B are independent events, then P (A and B) = P (A) · P (B). 3 Chapters 14-15 More probability rules • sample space set of all possible outcomes of a probabilistic event • Venn diagram diagram of the sample space of an event (represented by a rectangle) that depicts the relations among various collections of outcomes (represented by circles which might overlap) • General Addition Rule If A and B are any two events, then P (A or B) = P (A) + P (B) − P (A and B). 4 Chapters 14-15 • conditional probability If A and B are any two events, then the conditional probability P (B|A) of “event B given event A” is the frequency of the outcomes in B conditioned by the outcomes in A; that is, (rel.) freq. of outcomes in B also in A P (B|A) = (rel.) freq. of outcomes in A This is equivalent to the formula: P (A and B) P (B|A) = . P (A) • General Multiplication Rule If A and B are any two events, then P (A and B) = P (A) · P (B|A). • Events A and B are independent precisely when their conditional probabilities are the same as their unconditional probabilities; that is, when either one (and thus both) of these formulas hold: P (B|A) = P (B), P (A|B) = P (A). 5 Chapters 14-15 • tree diagram a diagram of the outcomes of pairs of successive events, in which the first level of branches represent outcomes of one event and the second layer outcomes of the second; useful for working with conditional probabilities • Bayes’ Rule Describes how to find the conditional probability of a pair of events when the conditions are reversed: if we know the conditional probability P (B|A), then we can find P (A|B) using a tree diagram, or via Bayes formula: P (A|B) = P (B|A)P (A) . C C P (B|A)P (A) + P (B|A )P (A ) 6