The collection

Every essay — page 24

Page 24 of 44, 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.

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

6 figures · new

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

6 figures · new

Number

The whole numbers, and how much structure they turn out to have.

The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them.

The primes are what is left over

Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

There is no last prime

7 figures $φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.$

A fraction that never closes

The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants.

Every fraction, exactly once

7 figures $8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.$

How close a fraction can get

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

The shape of a number's divisors

28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

Numbers that are their own parts

The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Two squares, and a lattice

Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Necklaces of 5 beads in 2 colours. Every string of beads, grouped by the rotations that carry one onto another.

Necklaces that prove a theorem

One number, two dials: 3 and 5. A grid of remainder pairs, each cell holding the smallest number that leaves those two remainders.

Two dials at once