week4class - Randomness and Probability Models Randomness...

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Randomness and Probability Models Randomness and Probability A phenomenon is random if individual outcomes are uncertain, but a regular distribution of outcomes emerges with a large number of repetitions. Example: The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of independent repetitions, Probability is a long-term relative frequency.
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Probability Models Probability models attempt to model random behavior. Consist of two parts: The probability of an event A, denoted by P(A), can be considered as the long run relative frequency of the event A. Sample Space and Events Sample space S : the set of all possible outcomes of a random phenomenon. Example: one Free throw attempt (FTA). A sequence of two FTA. Record all possible outcomes . Two FTA. Record the number of possible points scored in the two FTA. Event: An outcome or a set of outcomes of a random phenomenon, i.e. a subset of the sample space.
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Sample Space a sample space of a random experiment is the set of all possible outcomes. Simple events The individual outcomes are called simple events. Event An event is any collection of one or more simple events Our objective is to determine P( A ), the probability that event A will occur.
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Sequence of three free - throw attempts: There are 8 simple events, among which are . Some compound events include Boy or girl? An experiment in a hospital consists of recording the gender of each newborn infant until the birth of a male is observed. The sample space of this experiment is
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Basic Concepts in Probability The union of two events A and B , A B , is – The complement of an event A , A C , is – The intersection of two events A and B , A B , is –
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When two evens A and B have no outcomes in common, they are said to be - Venn Diagram Example Let A = { 0, 2, 4, 6, 8, 10}, B = { 1, 3, 5, 7, 9}, C = {0, 1, 2, 3, 4, 5}. A B = A B C C = A C = .
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Axioms of Probability Axiom 1: Axiom 2: Axiom 3: Properties of probability:
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Roll a fair die once Sample Space: S = {1, 2, 3, 4, 5, 6}. Each simple event is equally likely to occur. So P(1) = P(2) = … = P(6) = 1/6. Consider the following events. A: The number observed is at most 2. B: The number observed is an even number. C: The number 4 turns up. Find P(A) = P(A C ) = P(A and B) = P(A or C) = P(A or B) = 1 2 4 5 6 3
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Example: In a group of 88 people in Stat 10, 11 out of 50 women and 8 out of 38 men wear glasses. What is the probability that a person chosen at random from the group is a woman or someone who wears glasses? Example:
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week4class - Randomness and Probability Models Randomness...

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