The collection

Every essay — page 26

Page 26 of 49, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Probability

Randomness with a shape.

Tails of a sum of rare events: exact, and three bounds. 10: exact 5.42e-1, Bennett 1.00e+0, Bernstein 1.00e+0, Hoeffding 1.00e+0; 15: exact 8.34e-2, Bennett 3.39e-1, Bernstein 3.42e-1, Hoeffding 9.95e-1; 20: exact 3.44e-3, Bennett 2.09e-2, Bernstein 2.35e-2, Hoeffding 9.80e-1; 25: exact 4.64e-5, Bennett 3.66e-4, Bernstein 5.50e-4, Hoeffding 9.56e-1; 30: exact 2.46e-7, Bennett 2.34e-6, Bernstein 6.10e-6, Hoeffding 9.23e-1; 35: exact 5.88e-10, Bennett 6.45e-9, Bernstein 3.92e-8, Hoeffding 8.82e-1; 40: exact 7.03e-13, Bennett 8.70e-12, Bernstein 1.67e-10, Hoeffding 8.35e-1; 45: exact 4.56e-16, Bennett 6.27e-15, Bernstein 5.21e-13, Hoeffding 7.83e-1; 50: exact 1.71e-19, Bennett 2.59e-18, Bernstein 1.27e-15, Hoeffding 7.26e-1.

Charged for the variance, not the range

How long x² + c runs before repeating, for four constants. Four survival staircases on a logarithmic scale of length: two following the birthday curve, and two for c equal to 0 and minus 2 lying far to the right.

Two constants that do not walk at random

A single table for 9 guests over 4 nights. Small circles of 9 guests, one per night, each showing the night's seating as a closed zigzag path; every pair of guests is adjacent in exactly one of them.

Every pair side by side, once

A fair coin judged by someone who thinks it is biased. Posterior probabilities of three wrong coin biases over two thousand tosses of a fair coin, for five runs; the bias 0.6 wins every run.

Certain of a coin that is not there

Walks that prefer the steps they have taken. 12 Pólya-urn walks of 2000 steps, each heading off along its own straight line, with final speeds 0.99, −0.87, −0.02, −0.66, 0.01, 0.78, 0.47, 0.44, −0.04, 0.38, 0.83, 0.55, against the ±2√n band of an ordinary walk.

A walk that follows its own footsteps

Five urns, each drawn as walks. Simulated walks of 2000 draws for urns with replacement matrices (0,1,1,0), (2,1,1,2), (3,1,1,3), (7,1,1,7), (1,0,0,1).

An urn forgets its start only below one half

Where a jump puts the error of evenly spread points. Three unit squares with 16, 64, 256 scrambled Sobol' points, one per box, and the line y = 0.15 + 0.7x crossing 6, 12, 26 boxes.

What a jump costs evenly spread points

Leading digits of products of random numbers, against Benford's law. 1 factor: ones 11.1%, nines 11.1%; 2 factors: ones 24.1%, nines 3.4%; 3 factors: ones 30.1%, nines 4.2%; 6 factors: ones 30.1%, nines 4.6%; the law gives 30.1% and 4.6%.

Multiplying makes the digit one common

Four lists, merged in pairs, find a cancelling quadruple. A merge tree for four lists of 256 random 24-bit strings: two merges on the low 8 bits, then one on all 24, ending in 1 solution(s).

Four lists find a collision sooner

One shared birthday, as a measurement of the year. Histogram over 20000 rooms of the estimate T(T − 1)/2 from the first shared birthday, averaging 361 against 365, spread widely.

Counting a population by its repeats

Records in a thousand draws, and the lengthening waits between them. A sequence of 1000 uniform draws on a logarithmic position axis, with records at positions 1, 6, 11, 27, 36, 145, 539, 927.

A wait that ends and has no average

Running totals of random steps, stopped as they pass 4. Four staircases of uniform steps stopped on passing 4, taking 11, 8, 8, 10 steps; the exact mean is 8.666604.

Two-thirds of a step past the line

The longest run of heads at three sizes, never the same shape. n=65536: 12:0.018 13:0.117 14:0.233 15:0.239 16:0.172 17:0.104 18:0.057 19:0.030 20:0.015; n=82273: 13:0.075 14:0.204 15:0.249 16:0.197 17:0.124 18:0.070 19:0.037 20:0.019; n=104408: 13:0.040 14:0.162 15:0.248 16:0.221 17:0.148 18:0.086 19:0.046 20:0.024 21:0.012.

The longest run has no limit

A share that depends on the average

Two edges of a small graph, and whether one makes the other less likely. tree: 8 total, 5 with e, 5 with f, 3 with both, ratio 0.9600; forest: 24 total, 10 with e, 10 with f, 4 with both, ratio 0.9600; connected: 14 total, 10 with e, 10 with f, 7 with both, ratio 0.9800.

Edges that crowd each other out

Two random strings of bits and the longest thread between them. Two random binary strings of length 36; longest common subsequence 27.

The longest thread between two random strings

A fixed p-value favours the null more as the sample grows. Even odds at p=0.05 near n=76, at p=0.01 near n=1202, at p=0.001 near n=83176; p=0.05 at n=10⁶ gives 116.9 to one for the fair coin; PEAR data z=3.614, p=0.00030, BF₀₁=11.90.

A result at five per cent that favours the null

Peeking until the answer is yes

Predicting the unseen works until the future outgrows the past. Zipf population of 5000, mean sample 2000: seen 756, f1 561, f2 95; at t = 1, 2, 4: truth 489/873/1462, smoothed 492/843/1243.

How many new kinds the next sample will show

Two losing games that win together. A alone: -3.000 after 300; B alone: -3.131 after 300; A or B at random: 4.428 after 300; A A B B repeated: 4.322 after 300.

Two losing games that win together

The chance a random formula can be satisfied. n=20: 3:1.000 3.25:0.988 3.5:0.988 3.75:0.938 4:0.863 4.25:0.762 4.5:0.563 4.75:0.400 5:0.200 5.25:0.275 5.5:0.100 5.75:0.050 6:0.013; n=40: 3:1.000 3.25:1.000 3.5:1.000 3.75:0.963 4:0.863 4.25:0.575 4.5:0.425 4.75:0.138 5:0.050 5.25:0.025 5.5:0.037 5.75:0.000 6:0.000; n=80: 3:1.000 3.25:1.000 3.5:1.000 3.75:1.000 4:0.912 4.25:0.613 4.5:0.275 4.75:0.075 5:0.050 5.25:0.000 5.5:0.000 5.75:0.000 6:0.000.

A threshold nobody can place

6 figures $Four fractions staked on one run of tosses. p=0.6, 1000 tosses, 604 heads; final log10 wealth: Kelly 9.449, half Kelly 6.881, twice Kelly 0.409, everything ruined at toss 10.$

The best average is the worst bet

Knowing the coin, learning it, and guessing it once. p=0.6, 2000 tosses; final log10 wealth: hindsight 19.30, knew p 19.25, learned 17.73, plug-in from 20 tosses (15 heads, f=0.50) -25.88.

What not knowing the coin costs

Twelve studies of one coin, three ways of adding them up. theta=0.6, 12 studies of 40: heads 26,21,25,25,25,22,25,20,23,34,22,27; log10 learned product 0.06,-0.33,0.10,0.56,1.05,1.00,1.49,1.13,1.30,3.42,3.35,4.27; fresh 0.06,-0.63,-0.80,-0.98,-1.16,-1.78,-1.96,-2.67,-3.19,0.66,0.03,0.38; Fisher p 8.07e-2,2.58e-1,1.71e-1,1.19e-1,8.54e-2,1.34e-1,9.71e-2,1.72e-1,1.96e-1,7.44e-4,1.40e-3,4.71e-4; cross at 7 (learned e) and 10 (p).

Evidence that can be multiplied