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Bayes — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.

    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.

    part 1 · probability
  2. 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.

    part 2 · probability
  3. 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.

    part 3 · probability
  4. 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.

    part 4 · probability

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