Matpic
mathematics, in pictures
Concept

Likelihood

The probability of the observed evidence under a given hypothesis, read as a measure of how well that hypothesis explains it. In Bayes' theorem the ratio of two likelihoods is the factor by which the evidence multiplies the odds between the hypotheses.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Three doors, as areas. Staying wins 33.3% of the time and switching wins 66.7%, because the host's choice is constrained by what the host can see, so opening a door rules a region out without moving any boundary.

The door that was not opened

Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.

probability · Bayes Two children, and at least one is a boy. Four equally likely families drawn as quarters of a square: the question “is at least one a boy?” rules out only the girl–girl family, and leaves three equal quarters; the chance of two boys is 33.3%.

Two children and the sentence about one of them

A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.

probability · Bayes One experiment, measured by runs and by awakenings. Two unit squares for the same coin and the same schedule: the same experiment weighed two ways: by runs, heads keeps half the square; by awakenings, heads is one of 3 equal slices.

One coin, counted by runs and by wakings

Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.

probability · Bayes

Named alongside it

The objects these essays reach for when they reach for this one.

AreaBayes' theoremConditional probabilityCounting argumentSample spaceIndependenceLaw of large numbersExpectationMonty hall

All concepts