Mar 17, 2024 - Games, Sports & Leisure    Comments Off on Dominoes Solitaire

Dominoes Solitaire

This is a little game I invented to relieve boredom while spending Christmas in Berkeley. It has the feel of traditional dominoes, but can be played by one person. The challenge lies in planning the selection of tiles in advance so as to place as many tiles as possible. Since some tiles are not visible at the beginning of the game, the entire sequence cannot be planned, and therefore, some risk is involved.

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Nov 14, 2021 - Numbers    Comments Off on Percent Doubling

Percent Doubling

Expressing the rate of increase or decrease of a quantity as a relative value has been recognized as a challenging proposition for a long time because a consistent relative increase results in compounding effects (think compound interest). The quantity that is commonly used to express such changes, the percent difference, has some serious shortcomings when combining relative changes or when comparing changes over different periods. Here I propose an alternative way of expressing such changes, which is similar to the percent difference but which overcomes these shortcomings.

Plot of percent doubling vs. percent change

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Dec 2, 2020 - Games, Sports & Leisure    Comments Off on Blackjack Solitaire

Blackjack Solitaire

Blackjack, like most casino games, is a single-person game. It pits the player against the house, represented by the dealer, whose actions are controlled completely by the rules of the game. Here I present a way to play Blackjack as a game of solitaire.

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Apr 15, 2018 - Paradoxes, Probability    Comments Off on Boy or Girl Explained

Boy or Girl Explained

The “Boy or Girl Paradox” (also called “The Two Child Problem” in addition to other names) is generally phrased as follows:

You know a couple who has two children. At least one of the children is a girl. What is the probability that they have two girls?

This is an ambiguous problem, which leads to different answers depending on the assumptions that are used. Not enough information has been provided to produce a definite answer, and the unstated assumptions fill in the space needed to complete the logic.

Here I investigate this problem and explain the ambiguity.

[Boy or Girl?]
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Nov 18, 2016 - Education, Politics    Comments Off on Fearing for the Future

Fearing for the Future

While this report doesn’t explain the results of the recent election, it might explain the reaction to the election that we’re seeing at colleges and universities nationwide. This goes beyond just “safe spaces” and “trigger warnings.” Offering majors in basket weaving and “[fill-in-the-blank] studies” is doing nothing for the competitiveness of the US in the world market of ideas.

Think about this the next time someone (particularly a millennial) tries to tell you that “educated people” think this or think that. This report indicates that they don’t have the proper analytical skills to have an informed opinion (compared to most of their international peers or to previous generations), even the ones with the highest degrees.

Pay particular attention to the part that addresses “years of schooling” and “conferring of credentials and certificates.” Anyone who thinks that Sanders’s plan for “free college” would to do anything to help the country or the economy is fooling himself.
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Sep 1, 2015 - Numbers    Comments Off on Number Magic – An Explanation of Bendford’s Law

Number Magic – An Explanation of Bendford’s Law

Earlier this year, a colleague of mine sent me an email on Bendford’s Law. He had run across it somewhere and was fascinated by it. It seems counterintuitive that small digits would occur more frequently in the leading digits of arbitrary numerical data. One is tempted to think that arbitrary data would be made up of arbitrary digits, but that turns out not to be the case. It’s a genuine numerical phenomenon, and below I have provided a couple of ways to explain it. I point out that, utlimately, this law results from the notation that we use to represent real values.

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Aug 30, 2015 - Paradoxes, Probability, Statistics    Comments Off on Sleeping Beauty Plays the Lottery

Sleeping Beauty Plays the Lottery

I’ve already examined the classic Sleeping Beauty Problem and pointed out some of the pitfalls that many people fail to avoid when trying to solve the problem. I also examined Nick Bostrom’s so-called “Extreme Beauty” modification to the problem, in which Beauty wakes many, many times if the coin toss comes up tails. However, there is another “extreme” variant of this problem, the variant in which the coin toss is replaced with another two-result random process that has extremely uneven odds. That is, in this “extreme” problem, one of the possible results is extremely unlikely. Examining this variant with the methods of reasoning commonly used by the “thirders” can be enlightening and can provide some illustration of why they are wrong.

Since many “thirders” seem to be fond of relying on betting analogies to reason through the problem and explain their arguments, a useful substitute for the coin toss is a lottery. A typical lottery provides a very small chance of winning accompanied by a very large payoff (which is why lotteries are so popular). So here we shall examine what happens when Sleeping Beauty plays the lottery.

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Sleeping Beauty

For over 15 years, some people—particularly philosophers—continue to be confused by the so-called “Sleeping Beauty Problem.” This is a rather straight-forward exercise in conditional probability that should be accessible to a student in an undergraduate course on probability and statistics. Nevertheless, there are people who have managed to arrive at the wrong answer to this problem.

[Image from]
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