Mark Denny is the current DeNault Professor of Marine Sciences at Stanford University's Hopkins Marine Station in Pacific Grove, California. His books include "Air and Water: The Biology and Physics of Life's Media" and "Biology and the Mechanics of the Wave-Swept Environment" (both Princeton). Steven Gaines is Professor of Ecology, Evolution, and Marine Biology at the University of California, Santa Barbara.

"An excellent introduction to the uses of probability theory for a reader who is more familiar with biology than with mathematics ... Denny and Gaines have done a valuable service to biologists who are interested in a quantitative approach to life sciences."--Paul Janmey, Nature Cell Biology "A lively, well-written text... A student who reads this book closely will come away with a much deeper appreciation for the universality of diffusion mechanics in science, the deep connections between the distributions central to inferential statistics, the importance of extreme events and how to deal with them analytically, and, most importantly, the power and limitations inherent in the underpinning of the inferential statistics that the student has learned elsewhere."--Mark R. Patterson, American Scientist "This is a fantastic book. Indeed, one would be hard-pressed to find a more readable and lucid introduction to probability theory."--Gary B. Gillis, Journal of Experimental Biology

• Preface xi 1 The Nature of Chance 3 1.1 Silk, Strength, and Statistics 3 1.2 What Is Certain?7 1.3 Determinism versus Chance 8 1.4 Chaos 9 1.5 A Road Map 11 2 Rules of Disorder 12 2.1 Events, Experiments, and Outcomes 12 2.1.1 Sarcastic Fish 13 2.1.2 Bipolar Smut 14 2.1.3 Discrete versus Continuous 17 2.1.4 Drawing Pictures 18 2.2 Probability 19 2.3 Rules and Tools 20 2.3.1 Events Are the Sum of Their Parts 20 2.3.2 The Union of Sets 21 2.3.3 The Probability of a Union 23 2.3.4 Probability and the Intersection of Sets 24 2.3.5 The Complement of a Set 25 2.3.6 Additional Information and Conditional Probabilities 27 2.3.7 Bayes' Formula 29 2.3.8 AIDS and Bayes' Formula 30 2.3.9 The Independence of Sets 32 2.4 Probability Distributions 34 2.5 Summary 37 2.6 Problems 37 3 Discrete Patterns of Disorder 40 3.1 Random Variables 40 3.2 Expectations Defined 42 3.3 The Variance 46 3.4 The Trials of Bernoulli 48 3.5 Beyond 0's and 1's 50 3.6 Bernoulli = Binomial 51 3.6.1 Permutations and Combinations 53 3.7 Waiting Forever 60 3.8 Summary 65 3.9 Problems 66 4 Continuous Patterns of Disorder 68 4.1 The Uniform Distribution 69 4.1.1 The Cumulative Probability Distribution 70 4.1.2 The Probability Density Function 71 4.1.3 The Expectation 74 4.1.4 The Variance 76 4.2 The Shape of Distributions 77 4.3 The Normal Curve 79 4.4 Why Is the Normal Curve Normal?82 4.5 The Cumulative Normal Curve 84 4.6 The Standard Error 86 4.7 A Brief Detour to Statistics 89 4.8 Summary 92 4.9 Problems 93 4.10 Appendix 1: The Normal Distribution 94 4.11 Appendix 2: The Central Limit Theorem 98 5 Random Walks 106 5.1 The Motion of Molecules 106 5.2 Rules of a Random Walk 110 5.2.1 The Average 110 5.2.2 The Variance 112 5.2.3 Diffusive Speed 115 5.3 Diffusion and the Real World 115 5.4 A Digression on the Binomial Theorem 117 5.5 The Biology of Diffusion 119 5.6 Fick's Equation 123 5.7 A Use of Fick's Equation: Limits to Size 126 5.8 Receptors and Channels 130 5.9 Summary 136 5.10 Problems 137 6 More Random Walks 139 6.1 Diffusion to Capture 139 6.1.1 Two Absorbing Walls 142 6.1.2 One Reflecting Wall 144 6.2 Adrift at Sea: Turbulent Mixing of Plankton 145 6.3 Genetic Drift 148 6.3.1 A Genetic Diffusion Coefficient 149 6.3.2 Drift and Fixation 151 6.4 Genetic Drift and Irreproducible Pigs 154 6.5 The Biology of Elastic Materials 156 6.5.1 Elasticity Defined 156 6.5.2 Biological Rubbers 157 6.5.3 The Limits to Energy Storage 161 6.6 Random Walks in Three Dimensions 163 6.7 Random Protein Configurations 167 6.8 A Segue to Thermodynamics 169 6.9 Summary 173 6.10 Problems 173 7 The Statistics of Extremes 175 7.1 The Danger of Cocktail Parties 175 7.2 Calculating the Maximum 182 7.3 Mean and Modal Maxima 185 7.4 Ocean Waves 186 7.5 The Statistics of Extremes 189 7.6 Life and Death in Rhode Island 194 7.7 Play Ball! 196 7.8 A Note on Extrapolation 204 7.9 Summary 206 7.10 Problems 206 8 Noise and Perception 208 8.1 Noise Is Inevitable 208 8.2 Dim Lights and Fuzzy Images 212 8.3 The Poisson Distribution 213 8.4 Bayes' Formula and the Design of Rods 218 8.5 Designing Error-Free Rods 219 8.5.1 The Origin of Membrane Potentials 220 8.5.2 Membrane Potential in Rod Cells 222 8.6 Noise and Ion Channels 225 8.6.1 An Electrical Analog 226 8.6.2 Calculating the Membrane Voltage 227 8.6.3 Calculating the Size 229 8.7 Noise and Hearing 230 8.7.1 Fluctuations in Pressure 231 8.7.2 The Rate of Impact 232 8.7.3 Fluctuations in Velocity 233 8.7.4 Fluctuations in Momentum 235 8.7.5 The Standard Error of Pressure 235 8.7.6 Quantifying the Answer 236 8.8 The Rest of the Story 239 8.9 Stochastic Resonance 239 8.9.1 The Utility of Noise 239 8.9.2 Nonlinear Systems 242 8.9.3 The History of Stochastic Resonance 244 8.10 Summary 245 8.11 A Word at the End 246 8.12 A Problem 247 8.13 Appendix 248 9 The Answers 250 9.1 Chapter 2 250 9.2 Chapter 3 256 9.3 Chapter 4 262 9.4 Chapter 5 266 9.5 Chapter 6 269 9.6 Chapter 7 271 9.7 Chapter 8 273 Symbol Index 279 Author Index 284 Subject Index 286

Mark Denny is the current DeNault Professor of Marine Sciences at Stanford University's Hopkins Marine Station in Pacific Grove, California. His books include "Air and Water: The Biology and Physics of Life's Media" and "Biology and the Mechanics of the Wave-Swept Environment" (both Princeton). Steven Gaines is Professor of Ecology, Evolution, and Marine Biology at the University of California, Santa Barbara.

"An excellent introduction to the uses of probability theory for a reader who is more familiar with biology than with mathematics ... Denny and Gaines have done a valuable service to biologists who are interested in a quantitative approach to life sciences."--Paul Janmey, Nature Cell Biology "A lively, well-written text... A student who reads this book closely will come away with a much deeper appreciation for the universality of diffusion mechanics in science, the deep connections between the distributions central to inferential statistics, the importance of extreme events and how to deal with them analytically, and, most importantly, the power and limitations inherent in the underpinning of the inferential statistics that the student has learned elsewhere."--Mark R. Patterson, American Scientist "This is a fantastic book. Indeed, one would be hard-pressed to find a more readable and lucid introduction to probability theory."--Gary B. Gillis, Journal of Experimental Biology

• Preface xi 1 The Nature of Chance 3 1.1 Silk, Strength, and Statistics 3 1.2 What Is Certain?7 1.3 Determinism versus Chance 8 1.4 Chaos 9 1.5 A Road Map 11 2 Rules of Disorder 12 2.1 Events, Experiments, and Outcomes 12 2.1.1 Sarcastic Fish 13 2.1.2 Bipolar Smut 14 2.1.3 Discrete versus Continuous 17 2.1.4 Drawing Pictures 18 2.2 Probability 19 2.3 Rules and Tools 20 2.3.1 Events Are the Sum of Their Parts 20 2.3.2 The Union of Sets 21 2.3.3 The Probability of a Union 23 2.3.4 Probability and the Intersection of Sets 24 2.3.5 The Complement of a Set 25 2.3.6 Additional Information and Conditional Probabilities 27 2.3.7 Bayes' Formula 29 2.3.8 AIDS and Bayes' Formula 30 2.3.9 The Independence of Sets 32 2.4 Probability Distributions 34 2.5 Summary 37 2.6 Problems 37 3 Discrete Patterns of Disorder 40 3.1 Random Variables 40 3.2 Expectations Defined 42 3.3 The Variance 46 3.4 The Trials of Bernoulli 48 3.5 Beyond 0's and 1's 50 3.6 Bernoulli = Binomial 51 3.6.1 Permutations and Combinations 53 3.7 Waiting Forever 60 3.8 Summary 65 3.9 Problems 66 4 Continuous Patterns of Disorder 68 4.1 The Uniform Distribution 69 4.1.1 The Cumulative Probability Distribution 70 4.1.2 The Probability Density Function 71 4.1.3 The Expectation 74 4.1.4 The Variance 76 4.2 The Shape of Distributions 77 4.3 The Normal Curve 79 4.4 Why Is the Normal Curve Normal?82 4.5 The Cumulative Normal Curve 84 4.6 The Standard Error 86 4.7 A Brief Detour to Statistics 89 4.8 Summary 92 4.9 Problems 93 4.10 Appendix 1: The Normal Distribution 94 4.11 Appendix 2: The Central Limit Theorem 98 5 Random Walks 106 5.1 The Motion of Molecules 106 5.2 Rules of a Random Walk 110 5.2.1 The Average 110 5.2.2 The Variance 112 5.2.3 Diffusive Speed 115 5.3 Diffusion and the Real World 115 5.4 A Digression on the Binomial Theorem 117 5.5 The Biology of Diffusion 119 5.6 Fick's Equation 123 5.7 A Use of Fick's Equation: Limits to Size 126 5.8 Receptors and Channels 130 5.9 Summary 136 5.10 Problems 137 6 More Random Walks 139 6.1 Diffusion to Capture 139 6.1.1 Two Absorbing Walls 142 6.1.2 One Reflecting Wall 144 6.2 Adrift at Sea: Turbulent Mixing of Plankton 145 6.3 Genetic Drift 148 6.3.1 A Genetic Diffusion Coefficient 149 6.3.2 Drift and Fixation 151 6.4 Genetic Drift and Irreproducible Pigs 154 6.5 The Biology of Elastic Materials 156 6.5.1 Elasticity Defined 156 6.5.2 Biological Rubbers 157 6.5.3 The Limits to Energy Storage 161 6.6 Random Walks in Three Dimensions 163 6.7 Random Protein Configurations 167 6.8 A Segue to Thermodynamics 169 6.9 Summary 173 6.10 Problems 173 7 The Statistics of Extremes 175 7.1 The Danger of Cocktail Parties 175 7.2 Calculating the Maximum 182 7.3 Mean and Modal Maxima 185 7.4 Ocean Waves 186 7.5 The Statistics of Extremes 189 7.6 Life and Death in Rhode Island 194 7.7 Play Ball! 196 7.8 A Note on Extrapolation 204 7.9 Summary 206 7.10 Problems 206 8 Noise and Perception 208 8.1 Noise Is Inevitable 208 8.2 Dim Lights and Fuzzy Images 212 8.3 The Poisson Distribution 213 8.4 Bayes' Formula and the Design of Rods 218 8.5 Designing Error-Free Rods 219 8.5.1 The Origin of Membrane Potentials 220 8.5.2 Membrane Potential in Rod Cells 222 8.6 Noise and Ion Channels 225 8.6.1 An Electrical Analog 226 8.6.2 Calculating the Membrane Voltage 227 8.6.3 Calculating the Size 229 8.7 Noise and Hearing 230 8.7.1 Fluctuations in Pressure 231 8.7.2 The Rate of Impact 232 8.7.3 Fluctuations in Velocity 233 8.7.4 Fluctuations in Momentum 235 8.7.5 The Standard Error of Pressure 235 8.7.6 Quantifying the Answer 236 8.8 The Rest of the Story 239 8.9 Stochastic Resonance 239 8.9.1 The Utility of Noise 239 8.9.2 Nonlinear Systems 242 8.9.3 The History of Stochastic Resonance 244 8.10 Summary 245 8.11 A Word at the End 246 8.12 A Problem 247 8.13 Appendix 248 9 The Answers 250 9.1 Chapter 2 250 9.2 Chapter 3 256 9.3 Chapter 4 262 9.4 Chapter 5 266 9.5 Chapter 6 269 9.6 Chapter 7 271 9.7 Chapter 8 273 Symbol Index 279 Author Index 284 Subject Index 286