An elegant proof of Laplace’s rule of succession

Suppose I give you a bag of marbles and tell you that all the marbles are either green or black. You repeatedly reach in, pick out a random marble, and then put it back. You find that out of 100 draws, 70 of the marbles you took out of the bag were green. What's the probability that the next marble you'll draw will be green? Or, to put it another way, what's your best guess (expected value) of the fraction of marbles in the bag that are green?

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Chris the Criminal: A mathematical puzzle

(Warning: the comments on this post contain spoilers!) Chris the Criminal flips a fair coin. If it comes up heads he commits a crime. If it comes up tails, he doesn't. Now, Chris is being tried before a jury of n people (n is at least 2). The jury members all know the process by …

Scoring rules part 3: Incentivizing precision

[This is Part 3 of a three-part series on scoring rules. If you aren’t familiar with scoring rules, you should read Part 1 before reading this post. You don't need to read Part 2, but I think it's pretty cool.] In 9th grade I learned the difference between accuracy and precision from a classroom poster. …

Scoring rules part 2: Calibration does not imply classification

[This is Part 2 of a three-part series on scoring rules. If you aren't familiar with scoring rules (and Brier's quadratic scoring rule in particular), you should read Part 1 before reading this post. If you'd like, you can skip straight to Part 3.] One of the most important skills of good probabilistic forecasting is …

Scoring rules part 1: Eliciting truthful predictions

Yesterday I submitted for publication a paper I've been working on for a long time. The paper was on scoring rules, which I think are really interesting. In this three-part series, I'll tell you a bit about scoring rules and hopefully convey why I find them so cool. In this post I'll define scoring rules …

Pete’s problem

It's not every day that a presidential candidate stumps you with a really interesting math problem, but that's exactly what Pete Buttigieg did on Tuesday when he sent out the following email to his supporters: Pete’s Innovation Team handles data, engineering, and analytics responsibilities for the campaign, so you might not be shocked to learn …

Beyond the mean, median, and mode

[Thanks to Drake Thomas and Mike Winston for discussion.] In third grade math class, my teacher Ms. Potter taught my class about the mean, median, and mode of a list of numbers. What united these numbers, Ms. Potter told us, was that they were measures of central tendency: numbers that represented, in some sense, the …