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Ch. 2 - Notes - STATISTICS 321 Dr Soma Roy Chapter 2...

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STATISTICS 321 – Dr. Soma Roy Chapter 2: Probability Basic Ideas Probability theory is based on set theory. In an Experiment the outcome is subject to uncertainty. For example, A coin is tossed – it may land either “Heads” or “Tails.” A six–sided die is rolled – the number on top might be any of 1, 2, 3, 4, 5, 6. A person with a cold is given cold medicine – he may feel better or he may not. Sample Space , denoted by S , is the set of all possible outcomes of an experiment. For example, A six–sided die is rolled. List the outcomes of the sample space, S . A student is chosen at random. List the outcomes of the sample space, S for the answer to the following question: “What is the student’s blood type?” A machine produces nails. Due to variations in the production line, the length of the nails range from 0.9 inches to 1.1 inches. The sample space for the length of nails produced by this machine is Event is a subset of the sample space. For example, A six–sided die is rolled. List the outcomes that qualify for the event = even number showed on top . A machine produces nails. Due to variations in the production line, the length of the nails range from 0.9 inches to 1.1 inches. Event = nails at least an inch long. Combining Events Die Example : Roll a fair die, and record the outcome. Then, S = Let, A = event that an even number appears Qualifying outcomes = B = event that the number is 5 Qualifying outcomes = 1
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The Complement : of an event A is the event A c that contains all of the outcomes in S that are not in A. – Die Example : The Union : of 2 events A and B (denoted by A B) is the set containing all the outcomes that are either in A or B or both. ( Note: The key word is “or.”) – Die Example : The Intersection : of 2 events A and B (denoted by A B) is the set containing the outcomes that are common to A and B. ( Note: The key word is “and.”) – Die Example : The null event : is denoted by and is the empty set. That is, it contains no outcome. – Die Example : = event that the number is < 1. The 2 events C and D are mutually exclusive or disjoint if C D = . That is, when C and D have no outcomes in common. – Die Example : Axioms, Interpretations and Properties of Probabilities Given an experiment and a sample space S , probabilities are assigned to each event A, denoted by P ( A ), according to the following 3 rules: 1. Let S be the sample space. Then, 2
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2. For any event A , 3. If A and B are mutually exclusive events, More Probability Properties : Example: Sickle–cell and malaria A study in Africa tested 543 children for the sickle–cell trait and also for malaria infection. In all, 25% of the children had sickle–cell, 34.6% had malaria, and 6.6% had both.
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