February 15, 2017 Chapter 12: Introducing Probability 12.1 The Idea of Probability Probability is the science of chance behaviour Chance behaviour is unpredictable in the short run but has a regular and predictable pattern in the long run This is why we can use probability to gain useful results from random samples and randomized comparative experiments Random: individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions Relative frequency (proportion of occurrences) of an outcome settles down to one value over the long run. That one value is then defined to be the probability of that outcome. 12.2 The Search for Randomness Relative frequency probabilities o Can be determines (or checked) by observing a long series of independent trials (empirical data) Experience with many samples Simulation (computers, random number tables 12.3 Probability Models Descriptions of chance behaviour contain two parts: a list of possible outcomes and a probability for each outcome o The sample space S of a random phenomenon is the set of all possible outcomes o An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. o A probability model is a mathematical description of a random phenomenon consisting of 2 parts: a sample space S and a way of assigning probabilities to events. E.g. Probability model for flipping a coin o Sample space S = {Head, Tail} o Probability of heads = o Probability of tails = E.g. Probability model for tossing a single die o There are 6 possible outcomes: the sample space S = {1, 2, 3, 4, 5, 6} o Example event: the face that shown is even, S = {2, 4, 6} o Probability model: assign a number 16 to each outcome in a sample space E.g. Probability model for rolling 2 dice o Random phenomenon: roll a pair of fair dice o Sample space: (1,1), (1,2) (5,6), (6,6) o Probabilities: each individual outcome has a probability 136 (.0278) of occurring

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