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**Unformatted text preview: **MAT2378 Rafal Kulik Version 2009/Sep/30 Rafal Kulik MAT2378 Probability and Statistics for the Natural Sciences Chapter 3 Comments • These notes cover material from Chapter 3, Sections 3.3-3.5, 3.7-3.9. They are not complete, but summarize material delivered in class . For Chapter 3, I will use a blackboard. • I’m planning to spend 4 lectures on this material. • You may read Section 3.2 (random sampling). • Material from Section 3.6 will be given together with Chapter 4. Rafal Kulik 1 MAT2378 Probability and Statistics for the Natural Sciences Chapter 3 Recall from Chapter 2 We deal with random experiments (e.g. measurements of weight). In any “experiment” we obtain a sample (e.g. people in a particular hospital) and define the sample space as the set of all possible outcomes. This sample size is denoted by n . A sample space can be discrete or continuous . An event is a collection of outcomes from the sample space S . Events will be denoted by A , B , E 1 , E 2 etc. The complement of an event E , is the set of all outcomes which are not in E . Notation: E c . Rafal Kulik 2 MAT2378 Probability and Statistics for the Natural Sciences Chapter 3 Examples: • Toss a fair coin. The (discrete) sample space is { Head, Tail } . • Roll a die: The (discrete) s.s. is { 1 , 2 , 3 , 4 , 5 , 6 } . Various events: – Roll an even number: represent this as E = { 2 , 4 , 6 } . – Roll a prime number: E c = { 2 , 3 , 5 } . • Suppose we measure the weight (in grams) of a chemical sample. The (continuous) sample space can be represented by (0 , ∞ ), the pos. half line. Events: – sample is less than 1.5 grams: E = (0 , 1 . 5). Then E c = (1 . 5 , ∞ ); – sample exceeds 5 grams: (5 , ∞ ); Rafal Kulik 3 MAT2378 Probability and Statistics for the Natural Sciences Chapter 3 Classical Probability A probability is a numerical quantity that expresses the likelihood of an event. For situations where we have a random experiment which has exactly c possible disjoint , equally likely , simple outcomes we can assign a probability to an event E by counting the number of simple outcomes that correspond to E . If the count is a then P ( E ) = a c . if a sample space consists of N equally likely events, the probability of each outcome is 1 /N . Rafal Kulik 4 MAT2378 Probability and Statistics for the Natural Sciences Chapter 3 Examples: 1. Toss a fair coin. The sample space is { Head, Tail } The probability of observing a Head is 1 2 . 2. Throw a fair six sided die. There are 6 possible outcomes { 1 , 2 , 3 , 4 , 5 , 6 } . If E corresponds to observing a multiple of 3 then, in set notation E = { 3 , 6 } . Prob(number is a multiple of 3) = P ( E ) = 2 6 = 1 3 ....

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