# You're reading: Posts Tagged: probability

### Cushing your luck: properties of randomly chosen numbers

Long-time Aperiodical muse David Cushing has made a bet with us that he can give us an interesting post every Friday for the next ten weeks. Every week that he sends a post, we buy him a bar of chocolate. Every week that he doesn’t send us a post, he buys us a bar of chocolate. For his first trick, David is going to do some unnatural things with the natural numbers.

The greatest common divisor (gcd) of two or more integers is the greatest integer that evenly divides those integers. For example, the gcd of $8$ and $12$ is $4$ (usually written as $\gcd(8,12)=4$). Two integers are called coprime (or “relatively prime”) if their gcd is equal to $1$.

A reasonable question to ask is,

Given two randomly chosen integers $a$ and $b$, what is the probability that $\gcd(a,b)=1$?

### Ox Blocks probabilities

I have a new toy. ‘Ox Blocks’ box promises “Noughts and Crosses with a novel twist”.

### The Maths of Star Trek: The Original Series (Part I)

As you may well know, Star Trek was a science fiction TV show in the late 1960s. It featured futuristic technology and science fiction ideas such as warp drives, transporters, strange new worlds, time travel, and green alien space babes. And the possibility of all these things has, in the past, been discussed by experts, and nerds, in great detail. Especially that last one about green space babes.

But dammit, I’m a mathematician, not a physicist. So, instead of talking about the science of Star Trek yet again, what about the maths of Star Trek? After all, Star Trek is science fiction, but there is no such thing as maths fiction – so any mathematics featured on the show is sure to be on firmer ground. Right? Or as Spock himself says in ‘The Conscience of the King’;

SPOCK: Even in this corner of the galaxy, Captain, two plus two equals four.

Should we even expect much maths to feature on a simple space adventure show? In fact, many interesting mathematical ideas were raised during the show’s short run of 79 episodes, including; the probability we are alone in universe; a paradox that upset 20th century mathematicians as well as 23rd century androids; the mathematics of alien and Earth biology; and the most important question of all – when on a dangerous away mission, does the colour of your shirt really affect your chances of survival?

### Information and Inference: new journal with free content for two years

The Institute of Mathematics and its Applications has launched a new journal, Information and Inference: a Journal of the IMA. This aims to

publish high quality mathematically-oriented articles, furthering the understanding of the theory, methods of analysis, and algorithms for information and data.
Articles should be written in a way accessible to researchers in the associated topics in pure and applied mathematics, statistics, computer sciences, and electrical engineering. Articles are published in, but not limited to: information theory, statistical inference, network analysis, numerical analysis, learning theory, applied and computational harmonic analysis, probability, combinatorics, signal processing, and high-dimensional geometry.

The first paper, ‘The masked sample covariance estimator: an analysis using matrix concentration inequalities‘, has been made available for advanced online access.

More information: Oxford Journals: Information and Inference: a Journal of the IMA.

### The Table Never Lies

The table never lies, or so they say. So when Manchester City were crowned Premier League Champions last week everyone seemed to agree that they were the best team in the league. As Roberto Mancini said, they had scored more than United and conceded less and beaten them twice in the league. Although United finished on the same number of points it would be difficult to find a measure by which they deserved the title over City. Or would it?

### A student is artistic and loves poetry. Is it more likely she’s studying Chinese or Business?

Let’s suppose that:

1. $60\%$ of students who study Chinese are artistic and love poetry.
2. $20\%$ of students who study Business are artistic and love poetry, and
3. Only about $1\%$ of students study Chinese, whereas about $15\%$ of students study Business.

Thus, out of every $1000$ students, there are $10$ studying Chinese, of whom $6$ are artistic and love poetry, and also there are $150$ studying Business, of whom $30$ are artistic and love poetry.

So if a student is artistic and loves poetry, it’s $5$ times more likely she’s studying Business than Chinese.

So much for preconceptions (and “correlation”).

### The Odds Gods smile on birthday/card matches

The classic birthday problem asks how many people are required to ensure a greater than 50% chance of having at least one birthday match, meaning that two or more people share a birthday. The surprisingly small answer, assuming that all birthdays are equally likely and ignoring leap years like 2012, is 23 people.