Probability and Combinatorics
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1  Expectations
1.1  Ratio Populations	4
1.2  Differences Between Normal Samples	11
1.3  Biased and Anti-Biased Variance Estimates	13
1.4  Integrating the Bell Curve	16
1.5  The Expectation of Posterior Beliefs	24
1.6  Independence and Negation	28
1.7  Bernoulli Trials	30
1.8  Two-Dimensional Runs in Bernoulli Trials	32
2  Partitions
2.1  Computing the Partitions of n	37
2.2  Additive and Multiplicative Partitions	40
2.3  Partitions into Consecutive Integers	48
2.4  Partition Transform Cycles	50
2.5  Self-Descriptive Numbers and Auto-Valent Partitions	52
2.6  Partitions into Distinct Parts	54
2.7  Probabilities for Splitting Sets	57
2.8  On Euler’s Pentagonal Theorem	64
3  Randomness
3.1  Random High-Low Searching	74
3.2  Expected Area of Random Polygon In a Circle	78
3.3  Probability of a Random Triangle Being Acute	83
3.4  Choice Without Context?	90
3.5  Random Triangles	92
3.6  Generating Random Numbers In Order	94
3.7  Discrete Circular Distributions	95
3.8  N Points On Sphere all in One Hemisphere	100
3.9  A Random Walk Through π	109
3.10  Hail Encounters	111
4  Urns
4.1  Balls In Bins	115
4.2  Bayesian Marbles	118
4.3  Markov's Marbles	120
4.4  A Barrel of Colors and Null Outcomes	122
4.5  Colored Balls in Urns	125
4.6  Balls In Bins With Limited Capacity	128
4.7  Generalized Birthday Problem (N Items in M Bins)	131
4.8  Botches, Failures, and Successes	132
4.9  On Two Trays	134
5  Permutations
5.1  Cumulative Permutation Sequences	137
5.2  Derangements and Generalizations	138
5.3  Permutation Loops	142
5.4  Permutation Puzzles and Logic Problems	150
5.5  Eulerian Numbers	157
6  Collections
6.1  The Collector's Problem	165
6.2  Collecting k Complete Sets of Coupons	167
6.3  Dice Rolling A Given Sum	173
6.4  Round Robin Ties	177
7  Meetings
7.1  Meeting Probabilities	181
7.2  Meeting Combinations and Interval Graphs	188
7.3  Probability of Intersecting Intervals	195
7.4  Combining Probabilities	204
7.5  Witness Words	211
7.6  Probabilities With Renaming	215
7.7  Boolean Expansion as Linear Interpolation	216
7.8  The Shape of Coincidence	218
8  Gambling
8.1  A Two-Stage Lottery	222
8.2  The Gambler's Ruin	223
8.3  Reaching Ruin In Limited Turns	234
8.4  Never Say Never	240
9  Deals
9.1  At Least One Girl	244
9.2  What’s the Deal?	246
9.3  Two Envelopes	257
9.4  Ordered Envelopes	268
10  Monotone Functions
10.1  Generating the Monotone Boolean Functions	272
10.2  Dedekind’s Problem	274
10.3  Progress on Dedekind's Problem	276
11  Strings
11.1  The Statistics of Counterintelligence	281
11.2  Sequences of Characters in Random Strings	283
11.3  Bit-String Orbits Under Rotate-XOR	291
11.4  The e in Petersburg	303
12  Arrangements
12.1  Regular Polygonal Arrangements	310
12.2  Pythagorean Graphs	313
12.3  Two Proofs for a Five Point Star	317
12.4  Golf Tournaments, Polyhedra, and Latin Squares	319
12.5  The Boxer and the Pigeons	322
12.6  Tables of 1 to N with Equal Columns	323
12.7  Coloring The Edges of an Icosahedron	325
12.8  Uniform Squares	328
12.9  Christmas Ornaments	336
12.10  On 13 Bridges	338
13  Enumerations
13.1  Ringing of the Changes (The Shape of 4!)	345
13.2  Complete Projection Cubes	350
13.3  Polyomino Enumerations	358
13.4  The Determinants of 4x4 Magic Squares	360
13.5  414298141056 Quarto Draws Suffice!	361
14  Miscellaneous
14.1  Infinite Parallel Redundancy	367
14.2  A Conditional Protocol	368
14.3  The Coin in Three Fountains	370
14.4  And Then There Were None	371
14.5  Binary Games	375
14.6  f(f(x))=exp(x) Generates the Multinomials	376
14.7  Mnemonic Coins and Pattern Avoidance	377
14.8  Bernoulli Numbers and Harmonic Series	386
14.9  Two In the Shop	388
14.10  Benford's Law	389
14.11  Optimizing Your Wife	393