# chapter4 - Chapter 4 Probability Studying Randomness...

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Chapter 4 Probability: Studying Randomness

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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

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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

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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

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Example: Wars Begun by Year (1482-1939)
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