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Concept

Conditional probability

The chance of one event given that another is known to have happened.

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

has itdoes nottests positivetrue, and positive: 0.99%false, and positive: 4.95%so of the positives,16.7% really have it

Bayes' theorem is a picture of a square

A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.

probability · bayes the first pick was right — staying winsthe prize is behind door 2 — switching winsthe prize is behind door 3 — switching winsthe host knowsstay: 33.3%switch: 66.7%3 equally likely worlds,3 of them surviveno boundary moved:a region was ruled out

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 01234560.000.100.200.300.400.50how many are looked at and passed overchance of taking the best14.3%35.0%41.4%40.7%35.2%26.2%14.3%every one of the 5,040 orders, for every threshold: looking at 2 of 7 and then taking the first that beats them wins2,088 times, which is 41.4%taking the first one wins 14.3% and so does taking the last, because both amount to choosing without looking —the rule beats them by a factor of 2.90

When to stop looking

Candidates arrive one at a time in a random order. Each must be accepted or rejected on the spot, with no going back and no way to know what is still to come. The best possible rule is to look at about a third of them and then take the first one that beats everything seen — and it works about a third of the time, however many there are.

probability · optimal stopping

Named alongside it

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

AreaBayes' theoremCounting argumentBase rateDecision proceduree, the numberExpectationFalse positiveHarmonic seriesIndependenceIrrevocable decisionLikelihood

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