Matpic
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Series

Expectation — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Waiting for all 6 kinds. One bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.

    How long until every one turns up

    Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

    part 1 · probability
  2. Waiting for all 6 when they are not equally likely. One bar per kind giving the expected wait for that kind on its own, with the rarest much the tallest, and the expected wait for the whole collection printed above them.

    The one that hardly ever comes up

    Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.

    part 2 · probability
  3. How long 4 equally likely patterns take to appear. A bar for each of 4 patterns of 3 coin tosses giving the expected number of tosses before it first appears, with the lengths at which each pattern overlaps itself listed.

    Two patterns, one chance, different waits

    HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.

    part 3 · probability
  4. How much the first player can guarantee, as the coin's bias moves. A plot of the first player's best guaranteed winning chance in Penney's game against the probability of heads, a third on a fair coin and rising past one half only when the coin is heavily biased.

    A coin that lets the first player win

    On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.

    part 4 · probability
  5. Three patterns that beat one another in a circle: HHHT, TTHH, HTTH. Three coin-toss patterns at the corners of a triangle with arrows showing which beats which in two-way races, and each pattern's chance of winning when all three race.

    Three patterns in a circle

    Race three coin patterns at once and the gamblers' accounting still gives each one's chance of arriving first — one fairness equation per pattern. What it does not give is any way to read the three-way result off the two-way ones. HHHT, TTHH and HTTH beat one another in a circle, and HHH loses both its head-to-head races and still finishes ahead of one of the patterns that beat it.

    part 5 · probability

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