Glossary / Expected value
Mathematicians say you should "expect" 3.5 from a roll of a die. There is no such face, and no roll will ever show it. Yet this number runs casinos, insurers and lotteries. Let's see what it really means and when to trust it.
“In plain words” is hidden on all puzzles.
No roll gives 3.5. But the average of all rolls creeps right there. At first it jumps around, and after a thousand rolls it nearly sticks to the line.
The rolls axis is logarithmic: each mark is ten times further than the previous one.
“Rigorous” is hidden on all puzzles.
Multiply each possible result by its chance and add them all up. You get the average your result will approach if you repeat the same thing very many times.
1 · ⅙ + 2 · ⅙ + 3 · ⅙ + 4 · ⅙ + 5 · ⅙ + 6 · ⅙ = 21/6 = 3.5
A ticket costs 20 UAH. One ticket in 100,000 wins 1,000,000 UAH, the rest win nothing. The expected win: 1,000,000 · 1/100,000 = 10 UAH. So out of every 20 hryvnias the lottery pays back 10 on average. Buy some tickets and see how it looks from each side.
“Deeper” is hidden on all puzzles.
E[X + Y] = E[X] + E[Y]
This rule doesn't need independence, which is exactly why it is so strong. Take n people drawing their hats at random from a pile. Let Iᵢ = 1 if person i gets their own hat, otherwise 0. Each has a 1/n chance, so E[Iᵢ] = 1/n. The number of lucky ones is the sum I₁ + … + Iₙ, and its expected value is n · 1/n = 1. The events are dependent (if the first nine got their own hats, the tenth surely does too), but that doesn't stop the expected values from adding up.
E = 1 + ⅚ · E ⇒ E = 6
The first roll always happens. With chance ⅙ it is a six, and we're done. With chance ⅚ it isn't, and we're back at the start: E more rolls on average lie ahead, since the die remembers nothing. That gives the equation above. In general, an event with chance p takes 1/p tries on average.
σ² = E[X²] − (E[X])² = 91/6 − 3.5² ≈ 2.92 · σ ≈ 1.71 · spread of the average = σ / √n
A single roll misses 3.5 by about 1.71. The average of n rolls misses by √n times less: about 0.54 after 10 rolls, 0.17 after 100, 0.05 after 1,000. To become ten times more precise you need a hundred times more rolls. This is the law of large numbers, and it is why a casino stays calm and a single player doesn't.
The middle value: line everyone up in order, and half are below it, half above.
A millionaire walks into a bar: the average income shoots up, the median barely moves.
The sum of all values divided by how many there are: what everyone would get if it were shared equally.
Almost everyone has more legs than average: the average is a little under two.
A number from −1 to +1 showing how closely two quantities move together.
In summer both ice cream sales and drownings go up. Ice cream isn't to blame: the heat drives both.
A measure of spread: how far values typically stray from the mean. The standard deviation is its square root, in the same units.
Two towns with the same average temperature: one stays near +15 all year, the other swings from −20 to +50.
A bell-shaped distribution: most values sit near the mean, and the further away, the rarer.
Every ball on a Galton board falls at random, yet thousands of balls always build the same bell.
The more repetitions, the closer the share of an outcome gets to its true probability.
The share of heads creeps towards 50%, while the gap between heads and tails in coins usually grows.
The chance of an event given that something else is already known.
Almost all NBA players are tall, but almost all tall people are not NBA players.
Events are independent if knowing about one doesn't change the chances of the other.
After five heads in a row tails is no more likely: the coin remembers nothing.
How many times you multiply the base by itself to get a number: log₂ 8 = 3, because 2 · 2 · 2 = 8.
A sheet of paper folded in half 42 times would be thicker than the distance to the Moon.
A sample some people get into more easily than others, so it describes everyone badly.
In 1936 a magazine polled millions of Americans and still picked the wrong election winner.
How often chance alone would produce a gap at least as big as yours if there were really no difference.
When there is no effect at all, p below 0.05 turns up as often as p between 0.50 and 0.55.
A range around a measured number, built by a method that covers the true value in a chosen share of samples, such as 95%.
Of twenty honest 95% intervals, on average one misses the truth, and none of them knows whether it is the one.
The probability that a test notices a real effect of a given size. A power of 80% means the test misses one working effect in five.
A weak A/B test that does find an effect almost always exaggerates it: a real +10% shows up on the dashboard as +24%.
How far an average or a rate computed from a sample usually misses the truth: σ divided by the square root of the sample size.
Halving the error takes four times as much data.