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Playing Cards with Bayes' Theorem

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 Wizard 330 is a home-brewed variant of Wizard, a trick-taking card game. (Think Spades or Euchre. That sort of game.) While strategizing, an interesting problem arises that can be solved very cleanly with some probability. To understand the problem, you don't need to understand the (admittedly very complicated) details of the game. All you need to know is the following: Wizard 330 is played with a deck of 90 cards whose contents are unknown at the beginning of the game. The deck is a random subset of a larger library of 120 cards. In particular, at the beginning of the game, the deck could have anywhere from zero to eight  wizards , which are special, powerful cards. It is important for strategy to have a sense of how many wizards are in the deck. The game has 15 rounds, and during each round, some number of cards are revealed. The first round reveals $1\times 6$ random cards, the next reveals $2\times 6$ random cards, up to the final round which reveals all $15\times 6 = ...
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What's So Great About Polynomials?

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American math education is notorious for introducing mathematical concepts long before it introduces the reason to care about them. Case in point: polynomials. You know them. They're the functions that look like this:   $$f(x) = x^2 - x$$ or like this $$f(x) = -3x^3 + x^2 - \frac{1}{2} x - \pi.$$ You probably learned about these when you took Algebra. If you're like most students, you weren't immediately given a very good reason to care! In this post, I'll try to patch this hole by motivating polynomials for a student at a low level. They're not as unnatural as they might first appear; there is at least one good reason to care about polynomials!   Writing Down Functions is Hard In your math education, you've probably spent a lot of time staring at weird functions like this:   or this:     or this:   Drawing weird squiggly functions is easy. But what if you actually wanted to write one of these down ? Like, think about the function in the last picture he...
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Probabilities Are Less Real Than You Think

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 "Sir, the possibility of successfully navigating an asteroid field is approximately three thousand seven hundred and twenty to one!" -- C-3PO Probability statements tend to carry some of the gravitas and authority of mathematics. I claim that we tend to take these statements too seriously. Real world events do not have intrinsic probabilities. Let's start with an anecdote. A Story About A Big Number I once worked as a counselor at the Ross program, which is a summer camp for mathematically inclined high school kids. (If you are such a high school kid,  you should apply !) During the afternoons, the counselors would often collect around tables in the library to "grade," meaning we would all collaborate to distract each other from our grading duties. One "grading" afternoon, a counselor pulled a random book from the library shelf behind him. He found a checkout receipt inside with a transaction number printed on it. He laid it in the middle of th...
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