{"product_id":"the-probabilistic-method-isbn-9781119061953","title":"The Probabilistic Method","description":"\u003cp\u003e\u003cb\u003ePraise for the \u003ci\u003eThird Edition\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e“Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” \u003c\/b\u003e\u003cb\u003e- \u003ci\u003eMAA Reviews\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eMaintaining a standard of excellence that establishes \u003ci\u003eThe Probabilistic Method \u003c\/i\u003eas the leading reference on probabilistic methods in combinatorics, the \u003ci\u003eFourth Edition \u003c\/i\u003econtinues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics.\u003c\/p\u003e \u003cp\u003eEmphasizing the methodology and techniques that enable problem-solving, \u003ci\u003eThe Probabilistic Method, Fourth Edition \u003c\/i\u003ebegins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the \u003ci\u003eFourth Edition \u003c\/i\u003efeatures:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eAdditional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques\u003c\/li\u003e \u003cli\u003eNew coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits\u003c\/li\u003e \u003cli\u003eUpdated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eThe Probabilistic Method, Fourth Edition \u003c\/i\u003eis an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The \u003ci\u003eFourth Edition \u003c\/i\u003eis also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eNoga Alon, PhD,\u003c\/b\u003e is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal \u003ci\u003eRandom Structures and Algorithms\u003c\/i\u003e, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eJoel H. Spencer, PhD,\u003c\/b\u003e is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal \u003ci\u003eRandom Structures\u003c\/i\u003e \u003ci\u003eand Algorithms \u003c\/i\u003eand is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of \u003ci\u003eRamsey Theory, Second Edition\u003c\/i\u003e, also published by Wiley.\u003c\/p\u003e \u003cp\u003ePREFACE xiii\u003c\/p\u003e \u003cp\u003eACKNOWLEDGMENTS xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART I METHODS 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 The Basic Method 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 The Probabilistic Method, 3\u003c\/p\u003e \u003cp\u003e1.2 Graph Theory, 5\u003c\/p\u003e \u003cp\u003e1.3 Combinatorics, 9\u003c\/p\u003e \u003cp\u003e1.4 Combinatorial Number Theory, 11\u003c\/p\u003e \u003cp\u003e1.5 Disjoint Pairs, 12\u003c\/p\u003e \u003cp\u003e1.6 Independent Sets and List Coloring, 13\u003c\/p\u003e \u003cp\u003e1.7 Exercises, 16\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eThe Erd˝os–Ko–Rado Theorem, 18\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Linearity of Expectation 19\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Basics, 19\u003c\/p\u003e \u003cp\u003e2.2 Splitting Graphs, 20\u003c\/p\u003e \u003cp\u003e2.3 Two Quickies, 22\u003c\/p\u003e \u003cp\u003e2.4 Balancing Vectors, 23\u003c\/p\u003e \u003cp\u003e2.5 Unbalancing Lights, 25\u003c\/p\u003e \u003cp\u003e2.6 Without Coin Flips, 26\u003c\/p\u003e \u003cp\u003e2.7 Exercises, 27\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eBrégman’s Theorem, 29\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Alterations 31\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Ramsey Numbers, 31\u003c\/p\u003e \u003cp\u003e3.2 Independent Sets, 33\u003c\/p\u003e \u003cp\u003e3.3 Combinatorial Geometry, 34\u003c\/p\u003e \u003cp\u003e3.4 Packing, 35\u003c\/p\u003e \u003cp\u003e3.5 Greedy Coloring, 36\u003c\/p\u003e \u003cp\u003e3.6 Continuous Time, 38\u003c\/p\u003e \u003cp\u003e3.7 Exercises, 41\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eHigh Girth and High Chromatic Number, 43\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 The Second Moment 45\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Basics, 45\u003c\/p\u003e \u003cp\u003e4.2 Number Theory, 46\u003c\/p\u003e \u003cp\u003e4.3 More Basics, 49\u003c\/p\u003e \u003cp\u003e4.4 Random Graphs, 51\u003c\/p\u003e \u003cp\u003e4.5 Clique Number, 55\u003c\/p\u003e \u003cp\u003e4.6 Distinct Sums, 57\u003c\/p\u003e \u003cp\u003e4.7 The Rödl nibble, 58\u003c\/p\u003e \u003cp\u003e4.8 Exercises, 64\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eHamiltonian Paths, 65\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 The Local Lemma 69\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 The Lemma, 69\u003c\/p\u003e \u003cp\u003e5.2 Property B and Multicolored Sets of Real Numbers, 72\u003c\/p\u003e \u003cp\u003e5.3 Lower Bounds for Ramsey Numbers, 73\u003c\/p\u003e \u003cp\u003e5.4 A Geometric Result, 75\u003c\/p\u003e \u003cp\u003e5.5 The Linear Arboricity of Graphs, 76\u003c\/p\u003e \u003cp\u003e5.6 Latin Transversals, 80\u003c\/p\u003e \u003cp\u003e5.7 Moser’s Fix-It Algorithm, 81\u003c\/p\u003e \u003cp\u003e5.8 Exercises, 87\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eDirected Cycles, 88\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Correlation Inequalities 89\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 The Four Functions Theorem of Ahlswede and Daykin, 90\u003c\/p\u003e \u003cp\u003e6.2 The FKG Inequality, 93\u003c\/p\u003e \u003cp\u003e6.3 Monotone Properties, 94\u003c\/p\u003e \u003cp\u003e6.4 Linear Extensions of Partially Ordered Sets, 97\u003c\/p\u003e \u003cp\u003e6.5 Exercises, 99\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eTurán’s Theorem, 100\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Martingales and Tight Concentration 103\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Definitions, 103\u003c\/p\u003e \u003cp\u003e7.2 Large Deviations, 105\u003c\/p\u003e \u003cp\u003e7.3 Chromatic Number, 107\u003c\/p\u003e \u003cp\u003e7.4 Two General Settings, 109\u003c\/p\u003e \u003cp\u003e7.5 Four Illustrations, 113\u003c\/p\u003e \u003cp\u003e7.6 Talagrand’s Inequality, 116\u003c\/p\u003e \u003cp\u003e7.7 Applications of Talagrand’s Inequality, 119\u003c\/p\u003e \u003cp\u003e7.8 Kim–Vu Polynomial Concentration, 121\u003c\/p\u003e \u003cp\u003e7.9 Exercises, 123\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eWeierstrass Approximation Theorem, 124\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 The Poisson Paradigm 127\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 The Janson Inequalities, 127\u003c\/p\u003e \u003cp\u003e8.2 The Proofs, 129\u003c\/p\u003e \u003cp\u003e8.3 Brun’s Sieve, 132\u003c\/p\u003e \u003cp\u003e8.4 Large Deviations, 135\u003c\/p\u003e \u003cp\u003e8.5 Counting Extensions, 137\u003c\/p\u003e \u003cp\u003e8.6 Counting Representations, 139\u003c\/p\u003e \u003cp\u003e8.7 Further Inequalities, 142\u003c\/p\u003e \u003cp\u003e8.8 Exercises, 143\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eLocal Coloring, 144\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Quasirandomness 147\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The Quadratic Residue Tournaments, 148\u003c\/p\u003e \u003cp\u003e9.2 Eigenvalues and Expanders, 151\u003c\/p\u003e \u003cp\u003e9.3 Quasirandom Graphs, 157\u003c\/p\u003e \u003cp\u003e9.4 Szemerédi’s Regularity Lemma, 165\u003c\/p\u003e \u003cp\u003e9.5 Graphons, 170\u003c\/p\u003e \u003cp\u003e9.6 Exercises, 172\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eRandom Walks, 174\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART II TOPICS 177\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Random Graphs 179\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Subgraphs, 180\u003c\/p\u003e \u003cp\u003e10.2 Clique Number, 183\u003c\/p\u003e \u003cp\u003e10.3 Chromatic Number, 184\u003c\/p\u003e \u003cp\u003e10.4 Zero–One Laws, 186\u003c\/p\u003e \u003cp\u003e10.5 Exercises, 193\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eCounting Subgraphs, 195\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 The Erd˝os–Rényi Phase Transition 197\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 An Overview, 197\u003c\/p\u003e \u003cp\u003e11.2 Three Processes, 199\u003c\/p\u003e \u003cp\u003e11.3 The Galton–Watson Branching Process, 201\u003c\/p\u003e \u003cp\u003e11.4 Analysis of the Poisson Branching Process, 202\u003c\/p\u003e \u003cp\u003e11.5 The Graph Branching Model, 204\u003c\/p\u003e \u003cp\u003e11.6 The Graph and Poisson Processes Compared, 205\u003c\/p\u003e \u003cp\u003e11.7 The Parametrization Explained, 207\u003c\/p\u003e \u003cp\u003e11.8 The Subcritical Regions, 208\u003c\/p\u003e \u003cp\u003e11.9 The Supercritical Regimes, 209\u003c\/p\u003e \u003cp\u003e11.10 The Critical Window, 212\u003c\/p\u003e \u003cp\u003e11.11 Analogies to Classical Percolation Theory, 214\u003c\/p\u003e \u003cp\u003e11.12 Exercises, 219\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eLong paths in the supercritical regime, 220\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Circuit Complexity 223\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Preliminaries, 223\u003c\/p\u003e \u003cp\u003e12.2 Random Restrictions and Bounded-Depth Circuits, 225\u003c\/p\u003e \u003cp\u003e12.3 More on Bounded-Depth Circuits, 229\u003c\/p\u003e \u003cp\u003e12.4 Monotone Circuits, 232\u003c\/p\u003e \u003cp\u003e12.5 Formulae, 235\u003c\/p\u003e \u003cp\u003e12.6 Exercises, 236\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eMaximal Antichains, 237\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Discrepancy 239\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Basics, 239\u003c\/p\u003e \u003cp\u003e13.2 Six Standard Deviations Suffice, 241\u003c\/p\u003e \u003cp\u003e13.3 Linear and Hereditary Discrepancy, 245\u003c\/p\u003e \u003cp\u003e13.4 Lower Bounds, 248\u003c\/p\u003e \u003cp\u003e13.5 The Beck–Fiala Theorem, 250\u003c\/p\u003e \u003cp\u003e13.6 Exercises, 251\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eUnbalancing Lights, 253\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Geometry 255\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 The Greatest Angle Among Points in Euclidean Spaces, 256\u003c\/p\u003e \u003cp\u003e14.2 Empty Triangles Determined by Points in the Plane, 257\u003c\/p\u003e \u003cp\u003e14.3 Geometrical Realizations of Sign Matrices, 259\u003c\/p\u003e \u003cp\u003e14.4 𝜖-Nets and VC-Dimensions of Range Spaces, 261\u003c\/p\u003e \u003cp\u003e14.5 Dual Shatter Functions and Discrepancy, 266\u003c\/p\u003e \u003cp\u003e14.6 Exercises, 269\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eEfficient Packing, 270\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Codes, Games, and Entropy 273\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Codes, 273\u003c\/p\u003e \u003cp\u003e15.2 Liar Game, 276\u003c\/p\u003e \u003cp\u003e15.3 Tenure Game, 278\u003c\/p\u003e \u003cp\u003e15.4 Balancing Vector Game, 279\u003c\/p\u003e \u003cp\u003e15.5 Nonadaptive Algorithms, 281\u003c\/p\u003e \u003cp\u003e15.6 Half Liar Game, 282\u003c\/p\u003e \u003cp\u003e15.7 Entropy, 284\u003c\/p\u003e \u003cp\u003e15.8 Exercises, 289\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eAn Extremal Graph, 291\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Derandomization 293\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 The Method of Conditional Probabilities, 293\u003c\/p\u003e \u003cp\u003e16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297\u003c\/p\u003e \u003cp\u003e16.3 Exercises, 302\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eCrossing Numbers, Incidences, Sums and Products, 303\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Graph Property Testing 307\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Property Testing, 307\u003c\/p\u003e \u003cp\u003e17.2 Testing Colorability, 308\u003c\/p\u003e \u003cp\u003e17.3 Testing Triangle-Freeness, 312\u003c\/p\u003e \u003cp\u003e17.4 Characterizing the Testable Graph Properties, 314\u003c\/p\u003e \u003cp\u003e17.5 Exercises, 316\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eTurán Numbers and Dependent Random Choice, 317\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAppendix A Bounding of Large Deviations 321\u003c\/p\u003e \u003cp\u003eA.1 Chernoff Bounds, 321\u003c\/p\u003e \u003cp\u003eA.2 Lower Bounds, 330\u003c\/p\u003e \u003cp\u003eA.3 Exercises, 334\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eTriangle-Free Graphs Have Large Independence Numbers, 336\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAppendix B Paul Erd˝os 339\u003c\/p\u003e \u003cp\u003eB.1 Papers, 339\u003c\/p\u003e \u003cp\u003eB.2 Conjectures, 341\u003c\/p\u003e \u003cp\u003eB.3 On Erd˝os, 342\u003c\/p\u003e \u003cp\u003eB.4 Uncle Paul, 343\u003c\/p\u003e \u003cp\u003e\u003cb\u003e\u003ci\u003eThe Rich Get Richer, 346\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAppendix C Hints to Selected Exercises 349\u003c\/p\u003e \u003cp\u003eREFERENCES 355\u003c\/p\u003e \u003cp\u003eAUTHOR INDEX 367\u003c\/p\u003e \u003cp\u003eSUBJECT INDEX 371\u003c\/p\u003e \u003cp\u003e\"This is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics.\" (\u003ci\u003eSpringer Nature\u003c\/i\u003e, 2016)\u003c\/p\u003e \u003cp\u003e\u003cb\u003eNoga Alon, PhD,\u003c\/b\u003e is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal \u003ci\u003eRandom Structures and Algorithms\u003c\/i\u003e, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eJoel H. Spencer, PhD,\u003c\/b\u003e is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal \u003ci\u003eRandom Structures\u003c\/i\u003e \u003ci\u003eand Algorithms \u003c\/i\u003eand is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of \u003ci\u003eRamsey Theory, Second Edition\u003c\/i\u003e, also published by Wiley.\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePraise for the \u003ci\u003eThird Edition\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e“Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” \u003c\/b\u003e\u003cb\u003e- \u003ci\u003eMAA Reviews\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eMaintaining a standard of excellence that establishes \u003ci\u003eThe Probabilistic Method \u003c\/i\u003eas the leading reference on probabilistic methods in combinatorics, the \u003ci\u003eFourth Edition \u003c\/i\u003econtinues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics.\u003c\/p\u003e \u003cp\u003eEmphasizing the methodology and techniques that enable problem-solving, \u003ci\u003eThe Probabilistic Method, Fourth Edition \u003c\/i\u003ebegins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the \u003ci\u003eFourth Edition \u003c\/i\u003efeatures:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eAdditional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques\u003c\/li\u003e \u003cli\u003eNew coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits\u003c\/li\u003e \u003cli\u003eUpdated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eThe Probabilistic Method, Fourth Edition \u003c\/i\u003eis an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The \u003ci\u003eFourth Edition \u003c\/i\u003eis also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47990320890085,"sku":"NP9781119061953","price":98.5,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119061953.jpg?v=1761787344","url":"https:\/\/k12savings.com\/products\/the-probabilistic-method-isbn-9781119061953","provider":"K12savings","version":"1.0","type":"link"}