school of mathematical science and computing technology in csu. /ColorSpace /DeviceRGB /Length 24960 When an experiment is performed under these conditions, certain elementary events occur in different but … outline. Create stunning presentation online in just 3 steps. /BitsPerComponent 8 probability, Probability Theory - . Probability Spaces and Sigma-Algebras (PDF) 2: Extension Theorems: A Tool for Constructing Measures (PDF) 3: Random Variables and Distributions (PDF) 4: Integration (PDF) 5: More Integration and Expectation (PDF) 6: Laws of Large Numbers and Independence (PDF) 7: Sums of Random Variables (PDF) 8: Weak Laws and Moment-Generating and Characteristic Functions (PDF) 9: Borel-Cantelli and … probability. Getting yatzy … Sample Space Examples • Definition: the set of all possible outcomes of an experiment. • What is the complimentary event of the 50s event? /Filter /FlateDecode the idea of probability is empirical , that is probability is base on observation. Elementary Probability theory. • What is the event space of each event? mathematical foundations i: probability theory. Randomness: In a random process, we know what outcomes could happen but we don’t know which particular outcome will … stream -fyFE ���4�����@��X0���$� �+&�h -�~�"�gE�+~��H `\�t)qE\ n ����� why making a prediction that is borne out makes a theory more probably true. review of probability theory. foundations of statistic natural language processing. Elementary Probability Theory. /Filter /FlateDecode The sample space is the collection or totality of all possible outcomes of a conceptual experiment. Equal Probability Sampling - . Introduction to probability theory - . /Type /XObject /Filter /DCTDecode basic probability concepts: sample spaces and events, simple probability, and joint, Chapter Five - . • Example: Flipping a coin 3 times • Each flip is a group and each flip has 2 possible outcomes (n1 = n2 = n3 =2) • The number of possible outcomes is 2*2*2 = 8 • Outcomes = {HHH}, {HHT}, {HTH}, {HTT}, {THH}, {THT}, {TTH}, {TTT}, Combinations Rule • Used to select all the possible groups of size r from the sample space • Since the sample space has n outcomes, r ≤ n • Example: • 4 cards - A, K, Q, J • How many combinations can you have if you pick 2 cards (r=2) • AK, AQ, AJ, KQ, KJ, QJ, Permutations Rule • Used to select all the possible groups of size r from the sample space including the order of the elements • Since the sample space has n outcomes, r ≤ n • Example: • 4 cards - A, K, Q, J • How many combinations can you have if you pick 2 cards (r=2) • AK, AQ, AJ, KQ, KJ, QJ • KA, QA, JA, QK, JK, JQ, Hypergeometric Rule • The combination of the product rule and the combination rule • Since the sample space has n outcomes, r ≤ n • Example: • 2 sets of 4 cards (A, K, Q, J) and (1, 2, 3, 4) • How many combinations can you have if you pick 2 cards (r=2) from each set? xڥV]o�6}���ok��)J���@�~���M�>{Pd�&"�E'M�Ε�8i9[ۢ)��{ι���� course groups of probability and, 6.2 – Probability - . probability basics. of the probability theory to understand and quantify this notion. There is a coin which gives HEADS with probability ¼ and TAILS with probability … Randomized Monte Carlo Algorithm forapproximate median . 5 0 obj << Chapter 5 of the textbook Pages 145-164. key terms characterizing outcomes and, Statistical NLP: Lecture 4 - . review of probability theory: outcome. probability theory -, CHAPTER 3 Probability Theory - . A Tutorial on Probability Theory A;B A[B B A 0.0 0.2 0.6 0.7 1.0 1.0 Figure 1: Graphical representation of operations with events. 3 view of probability. endobj ���^��]u�\L����_,o1>bk�M��c������ ������7���P�k[����f1l�4~ ;}�p�"�Fk�Dջ�g��G�G-|���p���b�ok�Ϧz�kN�9f��:��>q�u&�) Chapter 1 lists basic properties of ﬁnite-length random walks, including space-time … lecture iii. Get powerful tools for managing your contents. i think that bieren’s discussion of the, Presentation 4 - . Probability itself is a big topic and here it is not possible to discuss each and everything. Mathematical Foundations - . �� • P(answer of 31) = 1 / 10 = 0.1 • What is the probability of getting a result in the 30s? 7. :H���,)��b� Elementary probability computations can to some extent be handled based on intuition and common sense. rong jin. The basic situation is an experiment whose outcome is unknown before it takes place e.g., a) coin tossing, b)throwingadie,c)choosingatrandomanumberfromN,d)choosingatrandoma number from (0,1). Because S is the union of all possible events, its probability is p(S) = 1 and we represent S as a square of side 1. 1\�)g�d��)�x弯 ��T%�_�kA�1�i&�� • P(rolling a 6) = 1 / 6 = 0.166667 • The number of elements in the event space (m) = 1 (i.e., a 6) • The number of elements in the sample space (n) = 6 (i.e., 1,2,3,4,5,or 6), Complicating Factors • What do we do when all outcomes are not equally likely? 인공지능연구실, Introduction to Probability Theory - . rules of probability. /Subtype /Image /Type /ObjStm Elementary Probability Theory Chapter 5 of the textbook Pages 145-164. chapter 9 the development of probability theory: pascal, bernoulli, and laplace. the number of elements) in the group being formed, Product Rule • Used to calculate all possible combinations available when selecting one member from each available group • Number of possible combinations = n1 * n2 …. • Answer = 36, A B UNION Probability Theorems • Addition Theorem • Rule of thumb: Union uses addition, Examples • Coin Flip: • P(heads) = 0.5 • P(tails) = 0.5 • P(heads ∩ tails) = 0 • Cards • P(heart) = 13/52 • P(king) = 4/52 • P(heart ∩ king) = 1/52, Probability Theorems • Complementation Theorem • Recall that P(S) = 1, Probability Theorems • Conditional Probability • Think of this as the probability of X given Y where both X and Y have their own probability • Intuition should tell you that this will hinge on the intersection of X and Y, Probability Theorems • Statistically Independent Events – the probability of an event remains the same despite the occurrence of another event • Example: The probability of a coin flip being heads is ½ regardless of what the last coin flip was • Based on conditional probability, A INTERSECTION B Probability Theorems • Multiplication Theorem • Rule of thumb: Intersections use multiplication, Statistically Independent Examples • 2 Coin Flips • A and B are the probability of getting heads • P(heads) = 1/2 • P(heads ∩heads) = P(A|B) * P(B) = ¼ • P(heads | heads on first flip) = P(heads ∩heads) / P(B) = (¼) / (½) = 1/2 • Draw 2 cards • P(heart) = 13/52 • P(king) = 4/52 • P(heart ∩ king) = 1/52 • P(heart | king) = P(heart ∩ king) / P(king) = (1/52) / (4/52) = 13/52 = 1/4 • P(king | heart) = P(heart ∩ king) / P(heart) = (1/52) / (13/52) = 4/52 = 1/13, Statistically Dependent Example • Probability of drawing 2 hearts • Drawing single cards from a complete deck would equate to: • P(A) = P(heart) = 13/52 • P(B) = P(heart) = 13/52 • P(A|B) = P(heart|heart on last draw) = 12/51 • Solution 1: imagine drawing 1 card and then the second: • (A ∩ B) = P(A|B) * P(B) = 0.589 • Solution 2: imagine drawing both cards at once • Remember P(event) = m/n • n = number of all combinations (full sample space) • m = number of possible combinations of 2 hearts (event space) • Both m and n are calculated using the combinations rule • m = n = • P(drawing 2 hearts) = 78/1326 = 0.589, © 2020 SlideServe | Powered By DigitalOfficePro, - - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -.

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