• 3 Posts
  • 34 Comments
Joined 11 months ago
cake
Cake day: August 6th, 2023

help-circle
  • Since this is everyone’s favorite example of telescoping sums, let’s do it another way just for giggles.

    Combinatorial proof

    The denominator is P(n+1, 2) which is the number of ways for 2 specified horses to finish 1st and second in an n+1 horse race. So imagine you’re racing against horses numbered {1, 2, 3, …}. Either you win, which has probability 0 in the limit, or there is a lowest numbered horse, n, that finishes ahead of you. The probability that you beat horses {1,2, … , n-1} but lose to n is (n-1)! / (n+1)! or P(n+1, 2) or 1/(n2+n), the nth term of the series. Summing these mutually exclusive cases exhausts all outcomes except the infinitesimal possibility that you win. Therefore the infinite sum is exactly 1.






  • The Gargoyle by Andrew Davidson

    It starts with the nameless narrator experiencing a near-fatal car crash due to his drug and alcohol induced hallucinations. During his lengthy recovery in the burn ward, he meets a psych patient who alleges their story begins 700 years ago in a German monastery. She tells him stories about their past in Germany, including how she obtained and translated a copy of Dante’s Inferno predating all known German translations. She tells other stories too, about a Japanese glass craftswoman, an Icelandic Viking, an Italian blacksmith, maybe some others, most of whom die young and tragically. She’s also a talented sculptor of stone gargoyles, a skill she allegedly learned from the narrator. The narrator suspects her stories are just the delusions of a schizophrenic, but can’t go back to his pre-accident life, so he agrees to go home with her to continue his recovery, and maybe learn a little more about her and why she’s taken an interest in him.

    I’m about 3/4 through it and impressed with how it’s written. Unfortunately, I never read the Divine Comedy, so I’m pretty sure I missed some things that a better educated reader would have recognized.



  • Once, I made an account for something that let me write my own security question and answer. I thought that was much better than the usual options and wrote something that cryptically referenced a difficult problem I once worked on. The answer could possibly be found online, but only to someone who properly understood the question. Later, when I needed to authenticate myself again, I got my security question. The answer isn’t something you typically memorize, but I knew what the prompt meant and how to work it out so I did so.

    But I was too slow. Apparently you had to answer within one minute. It took me about ten so it locked me out. Tech support helpfully reset my password after merely verifying my phone number and SSN which are probably known to thousands.



  • Not sure about MIchigan in particular, but other states have, in relatively recent history, given ballot access to presidential candidates who were unambiguously constitutionally ineligible for the office. It doesn’t make much sense to me either, but apparently neither the 14th amendment, nor any other federal law restricts who can run for president, merely who can hold the office if elected.



  • I see some correct solutions for the 50% case here already, so this reply is going for a perfect score within two tries.

    There are 16 ways to answer the quiz, one of which is correct. Assuming you don’t repeat your previous answers, two attempts give you a 2/16 or 1/8 chance that one of them is perfect.

    Now if you get feedback between your attempts, you should be able to do better. Let’s see by how much and break it into cases:

    1. Your first guess is already perfect. This happens 1/16 of the time. No further guessing is needed.

    2. Your first guess is 50% correct. This happens 3/8 of the time. Picking one of the unguessed answers improves your score to 100% 1/6 of the time.

    3. Your first guess is completely wrong. This happens 9/16 of the time. Picking different answers for both questions wins 1/9 of the time.

    So the overall chance of a perfect score is the weighted sum of these cases or 1/16 + (3/8 * 1/6) + (9/16 * 1/9) = 3/16.






  • I really wish IRV advocates would stop lying about things like:

    since voters can feel free to support them without fear of inadvertently helping a candidate they definitely don’t want to win.

    There is absolutely a spoiler effect in IRV, and it isn’t just theoretical – it happened in one of the elections the article praises as successful.

    Any election system works well with only two choices. IRV improves very slightly on plurality and works well with many choices, provided only two of them matter. But as soon as you get three competitive candidates, exactly the thing many election reformers want to see, really counterintuitive things start to happen.


  • Seems simpler for the good people of Wisconsin to just vote on a new law that says whatever they think is proper. Obstetric science has advanced somewhat since the time when Ignaz Semmelweis first proposed doctors washing their hands before delivering babies (especially if they’d just come form the cadaver lab), so some of the reasoning behind the 1849 law might be out of date.

    Unfortunately, that would require certain politicians to go on record about something that might be used against them if they later ran a national campaign, so better to let the court take the matter out of their hands and (mis-?)interpret an old law in a politically advantageous way.