chapter4 - Chapter 4 Probability: Studying Randomness...

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Unformatted text preview: Chapter 4 Probability: Studying Randomness Randomness and Probability • Random: Process where the outcome in a particular trial is not known in advance, although a distribution of outcomes may be known for a long series of repetitions • Probability: The proportion of time a particular outcome will occur in a long series of repetitions of a random process • Independence: When the outcome of one trial does not effect probailities of outcomes of subsequent trials Probability Models • Probability Model: – Listing of possible outcomes – Probability corresponding to each outcome • Sample Space ( S ): Set of all possible outcomes of a random process • Event: Outcome or set of outcomes of a random process (subset of S ) • Venn Diagram: Graphic description of a sample space and events Rules of Probability • The probability of an event A , denoted P(A) must lie between 0 and 1 (0 ≤ P(A) ≤ 1) • For the sample space S , P(S) =1 • Disjoint events have no common outcomes. For 2 disjoint events A and B , P(A or B ) = P(A) + P(B) • The complement of an event A is the event that A does not occur, denoted A c . P(A)+P(A c ) = 1 • The probability of any event A is the sum of the probabilities of the individual outcomes that make up the event when the sample space is finite Assigning Probabilities to Events • Assign probabilities to each individual outcome and add up probabilities of all outcomes comprising the event • When each outcome is equally likely, count the number of outcomes corresponding to the event and divide by the total number of outcomes • Multiplication Rule: A and B are independent events if knowledge that one occurred does not effect the probability the other has occurred. If A and B are independent, then P(A and B) = P(A)P(B) • Multiplication rule extends to any finite number of events Example - Casualties at Gettysburg • Results from Battle of Gettysburg North South North South Killed 3155 2592 0.0331 0.0334 Wounded 14525 12709 0.1523 0.1640 Captured/Missing 5365 12227 0.0563 0.1578 Safe Survival 72324 49972 0.7584 0.6448 Total 95369 77500 1.0000 1.0000 Counts Proportions Killed, Wounded, Captured/Missing are considered casualties, what is the probability a randomly selected Northern soldier was a casualty? A Southern soldier? Obtain the distribution across armies Random Variables • Random Variable (RV): Variable that takes on the value of a numeric outcome of a random process • Discrete RV: Can take on a finite (or countably infinite) set of possible outcomes • Probability Distribution: List of values a random variable can take on and their corresponding probabilities – Individual probabilities must lie between 0 and 1 – Probabilities sum to 1 • Notation: – Random variable: X – Values X can take on: x 1 , x 2 , …, x k – Probabilities: P ( X = x 1 ) = p 1 … P ( X = x k ) = p k Example: Wars Begun by Year (1482-1939) • Distribution of Numbers of wars started by year • X = # of wars stared in randomly selected year...
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This note was uploaded on 06/04/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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chapter4 - Chapter 4 Probability: Studying Randomness...

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