X Contents 31 Conditional Probability and Independence 257 Independence 259 Independent Repeated Trials 261 The Monty Hall Problem 262 Recap 263 Exercises 263 32 Random Variables 266 Random Variables as Events 267 Independent Random Variables 269 Recap 270 Exercises 270 33 Expectation 271 Linearity of Expectation 276 Product of Random Variables 279 Expected Value as a Measure of Centrality 282 Variance 283 Recap 287 Exercises 287 Chapter 6 Self Test 289 7 Number Theory 293 34 Dividing 293 Div and Mod 296 Recap 297 Exercises 297 35 Greatest Common Divisor 298 Calculating the ged 299 Correctness 301 How Fast? 302 An Important Theorem 304 Recap 307 Exercises 307 36 Modular Arithmetic 309 A New Context for Basic Operations 309 Modular Addition and Multiplication 310 Modular Subtraction 311 Modular Division 313 A Note on Notation 318 Recap 318 Exercises 318x Contents 7 31 Conditional Probability and Independence Independence 259 Independent Repeated Trials The Monty Hall Problem Recap 263 Exercises 263 261 262 32 Random Variables 266 Random Variables as Events Independent Random Variables Recap 270 Exercises 270 33 Expectation 271 Linearity of Expectation 276 267 269 Product of Random Variables 279 Expected Value as a Measure of Centrality 282 Variance 283 Recap 287 Exercises 287 Chapter 6 Self Test 289 Number Theory 293 34 Dividing 293 Div and Mod 296 Recap 297 Exercises 297 35 Greatest Common Divisor 298 Calculating the gcd 299 Correctness 301 How Fast? 302 An Important Theorem 304 Recap 307 Exercises 307 36 Modular Arithmetic 309 A New Context for Basic Operations 309 Modular Addition and Multiplication 310 Modular Subtraction 311 Modular Division 313 A Note on Notation 318 Recap 318 Exercises 318 257 fo