William Hoza's Bloghttps://williamhoza.com/blog/en-usNotation for nonempty intersectionhttps://williamhoza.com/blog/notation-for-nonempty-intersection/2022-06-18https://williamhoza.com/blog/notation-for-nonempty-intersection/There is no standard mathematical symbol for the intersection relation. You've probably encountered the symbol for the intersection *operation*. If A and B are sets, then "the intersection" of A and B, denoted A \cap B, is the set of all x such that x \in A and x \in B. It's the middle part of a Venn diagram. That's fine, but I'm talking about the intersection *relation*. We say that A "intersects" B if there exists some x such that x \in A and x \in B. Continue reading: <a href="https://williamhoza.com/blog/notation-for-nonempty-intersection/">https://williamhoza.com/blog/notation-for-nonempty-intersection/</a>Determining the direction of a coin's biashttps://williamhoza.com/blog/determining-direction-of-coins-bias/2020-04-07https://williamhoza.com/blog/determining-direction-of-coins-bias/Suppose you're playing a two-player game, like maybe ping pong or rock-paper-scissors. In each round, one player gets a point. It's possible to play any number of rounds, and nothing much changes from one round to the next. How long should you play if you want to figure out who is better at the game? Let's model each round as a biased coin toss. One player gets a point with probability 1/2 + eps and the other player gets a point with probability 1/2 - eps, independently each round, for some eps > 0. The goal is to determine which player is which. Continue reading: <a href="https://williamhoza.com/blog/determining-direction-of-coins-bias/">https://williamhoza.com/blog/determining-direction-of-coins-bias/</a>Economic theory of abortion policieshttps://williamhoza.com/blog/economic-theory-of-abortion-policies/2020-04-07https://williamhoza.com/blog/economic-theory-of-abortion-policies/I recently read an interesting book by economist Phillip Levine. In Sex and Consequences, Levine studies the impact of abortion policies. Realistically, how are abortion rates, birth rates, and pregnancy rates likely to be affected by these government interventions? The book addresses this question using both theoretical models and empirical evidence. The whole book is worth reading, but for this blog post I'm going to focus on the theoretical models. I'll provide some interactive visualizations and a little commentary. Continue reading: <a href="https://williamhoza.com/blog/economic-theory-of-abortion-policies/">https://williamhoza.com/blog/economic-theory-of-abortion-policies/</a>A three-player Pictionary varianthttps://williamhoza.com/blog/three-player-pictionary-variant/2020-04-07https://williamhoza.com/blog/three-player-pictionary-variant/Pictionary is a classic game in which players try to guess what other players are drawing. Quintessentially, Pictionary is played by two teams of two players (one teammate draws while the other teammate guesses.) Hasbro's official Pictionary rules also describe a "three-player" version of the game. In Hasbro's "three-player" game, one player is designated as the artist for the entire game; the other two players compete to win by guessing. Hasbro's game has some merits, but the poor artist can hardly be considered a "player" in the game. Hasbro's game is more like a two-player guessing game. In this blog post, we describe a Pictionary variant with no teams in which every player has a shot at winning. Our game is designed for three players, but it also works with more than three players. Continue reading: <a href="https://williamhoza.com/blog/three-player-pictionary-variant/">https://williamhoza.com/blog/three-player-pictionary-variant/</a>Top 10 blatant contradictions in mathhttps://williamhoza.com/blog/top-10-blatant-contradictions-in-math/2020-04-07https://williamhoza.com/blog/top-10-blatant-contradictions-in-math/Sorry, matheists, but every thinking person admits that abstract objects don't exist. Math is just a fairy tale invented by professors to control their grad students. Pathetically, the fairy tale isn't even internally consistent: Continue reading: <a href="https://williamhoza.com/blog/top-10-blatant-contradictions-in-math/">https://williamhoza.com/blog/top-10-blatant-contradictions-in-math/</a>A "can" of wormshttps://williamhoza.com/blog/can-of-worms/2020-04-07https://williamhoza.com/blog/can-of-worms/In "The Incompatibility of Free Will and Determinism", Peter van Inwagen argues that if the universe is deterministic, then free will does not exist. (He is silent about whether the universe is in fact deterministic and about whether free will in fact exists.) This is in contrast to the compatibilist position, which holds that free will and determinism are not contradictory. Briefly, van Inwagen's argument is that when an agent with free will performs some action, she (by definition of "free will") could have performed a different action. But in a deterministic universe, acting a different way requires either altering the past or violating the laws of physics. So van Inwagen concludes that the free agent could have either altered the past or violated the laws of physics. Finally, van Inwagen says that it is obvious that nobody can alter the past, and by definition of the phrase "law of physics", nobody can violate the laws of physics either. So our hypothetical free agent in a deterministic universe cannot exist. Continue reading: <a href="https://williamhoza.com/blog/can-of-worms/">https://williamhoza.com/blog/can-of-worms/</a>