math 144 chap2

math 144 chap2 - Chapter 2 Probability Probability theory...

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Unformatted text preview: Chapter 2 Probability Probability theory is a branch of mathematics dealing with chance phenomena. The origins of the subject date back to the Italian mathematician Cardano about 1550, and French math- ematicians Pascal and Fermat in 1654. The modern mathematical foundation for probability theory was laid down by Russian mathematician A. N. Kolmogorov (1903-1989). Examples: • How can gambling be a profitable business for a casino? • If you buy a lottery ticket every day for many years, how much will each ticket win on the average? • Give a test for the AIDS virus to the employees of a small company. What is the chance of at least one positive test even if all the employees are free of the virus? • What is the probability of drawing an ace from an ordinary deck of 52 playing cards? • What is the probability of getting a heads when flipping a coin 2.1 Sample Space and Events A statistic takes various values in each individual sample but there is nevertheless a regular distribution in a large number of repetitions. This is called a random phenomenon. A random phenomenon has outcomes that we cannot predict but that have a regular distribution in repetitions Basic concepts: • Experiment refers to any process of observation or measurement that (i) can be repeated, theoretically, an infinite number of times; (ii) has a well-defined set of possible outcomes. • Sample space denote by Ω or S : The set of all possible outcomes of an experiment 2-1 • Event: a subset of the sample space Example 2.1 Roll a pair of coins, one red and one blue , what is the sample space? Solution : Example 2.2 Describe the event B that the total number of points rolled with the pair of dice is 7 Solution : Let A and B be any two events defined over the sample space Ω (a) The intersection of A and B , denoted by A ∩ B , (A and B), is the event whose outcomes belong to both A and B (both A and B occur) (b) The union of A and B , denoted by A ∪ B , (A or B), is the event whose outcomes belong to either A or B (either A or B occurs) Example 2.3 Ω = { 1 , 2 , ..., 6 } . 2-2 (a) A = { 2 , 3 , 4 } , B = { 2 , 4 , 6 } . Then A ∩ B = { 2 , 4 } , A ∪ B = { 2 , 3 , 4 , 6 } (b) A = { 1 , 3 , 5 } , B = { 2 , 4 , 6 } . Then A ∪ B = { 1 , 2 , 3 , 4 , 5 , 6 } , A ∩ B = ∅ • A and B are said to be mutually exclusive (disjoint) if they have no outcomes in common, that is, A ∩ B = ∅ , null set, emptyset • The complement of A , denoted by A c or A , is the event consisting of all the outcomes in Ω other than those contained in A ( A does not occur) Remark: A and A c are disjoint, A ∪ A c = Ω • De Morgan Law: ( A ∪ B ) c = A c ∩ B c ( A ∩ B ) c = A c ∪ B c 2-3 2.2 Counting Sample Points 2.2.1 The Basic Principle of Counting If an operation consist of a sequence of k separate steps of which the first can be performed in n 1 ways, followed by the second n 2 ways, and so on until the k th can be performed in n k ways, then the operation consisting of...
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This note was uploaded on 09/24/2010 for the course MATH 144 taught by Professor Lisiufeng during the Fall '09 term at HKUST.

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math 144 chap2 - Chapter 2 Probability Probability theory...

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