01 - p. 1-1 Introduction to Probability •...

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Unformatted text preview: p. 1-1 Introduction to Probability • Uncertainty/Randomness ( 不確定性 / 隨機性 ) in our life Many events are random in that their result is unknowable before the event happens. Will it rain tomorrow? How many wins can 王建民 achieve this season? What numbers will I roll on two dice? Q : Is your height/weight measure random? We often want to assess how likely are the outcomes of such events. Probability is that measurement. • Distinction between Discovery ( 發現 ) and Invention ( 發明 ), e.g., 哥倫布“發現”新大陸 愛迪生“發明”電燈泡 Q : 相對論是發明還是發現? • 機率論是人類“發明”來處理生活中的不確定性之理論 愛因斯坦 : “ 上帝永遠不會擲骰子 ” p. 1-2 Combinatorial Analysis • An example: A communication system is to consist of n seemingly identical antennas that are to be lined up in a linear order A resulting system will be functional as long as no two consecutive antennas are defective If it turns out m (=2) of the n (=4) antennas are defective, what is the probability that the resulting system will be functional? • Many problems in probability theory can be solved simply by counting the number of different ways that a certain event can occur • The mathematical theory of counting is formally known as combinatorial analysis • What to Count? (i) Lists, (ii) Permutations, (iii) Combination, (iv) Partition, (v) Number of integer solutions. p. 1-3 • Lists Definition Order Pairs: ( x , y )=( w , z ) iff w = x and z = y . Ordered Triples: ( x , y , z )=( u , v , w ) iff u = x , v = y , and w = z . List of Length r : ( x 1 , …, x r )=( y 1 , …, y s ) iff s = r and x i = y i for i =1, …, r . Example (License Plates): A license plate has the form LMNwxyz , where L , M , N {A, B,…, Z}, w , x , y , z {0, 1,…, 9}, and, so, is a list of length seven....
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01 - p. 1-1 Introduction to Probability •...

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