131A_1_Lecture_2-1_Winter_2012

131A_1_Lecture_2-1_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 2 - 1
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UCLA EE131A (KY) 2 Lecture 2 -1: Learning how to count • For a discrete sample space with finite number of sample points, in order to evaluate the relative frequency f k = N k /N, of the k-th event of interest, we need to be able to count, N k , the number times the k-th event occurs divided by the total number of outcomes N. • Thus, we need to find a general methods of counting both the elements that make up N k as well as the elements that make up N . • Combinatorial analysis is the study of counting.
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UCLA EE131A (KY) 3 Motivations on the need for counting (1) • Let us consider some probability problems that will motivate us to learn how to “count.” • At this point, we can equate the probability of an event as the same as the relative frequency of that event. We will formalize this property later. Ex. 1 – Consider a box with two black (B) objects and one white (W) object as shown in Fig. 1. Fig. 1 A box with 2 B and 1 W objects
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UCLA EE131A (KY) 4 Motivations on the need for counting (2) Let us pick randomly (i.e., sample) one object from the box in Fig. 1. Let us find the probabilities of some events. Prob(“object is white”) = P(W) = Relative freq.(“object is white”) = RF(W) = 1/3 . (1) The number 3, in the denominator of (1), is due to the number of objects that we can pick in the box is 3. The number 1, in the numerator of (1), is due to the number of object in the box satisfying the event, “object is white” is 1.
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UCLA EE131A (KY) 5 Motivations on the need for counting (3) Similarly, Prob(“object is black”) = P(B) = Relative freq.(“object is black”) = RF(B) = 2/3 . (2) The number 3, in the denominator of (2), is due to the number of objects that we can pick in the box is 3. The number 2, in the numerator of (2), is due to the
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131A_1_Lecture_2-1_Winter_2012 - EE 131A Probability...

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