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

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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
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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
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(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
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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 k steps can be performed by n 1 × n 2 × · · · × n k ways .
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