You're reading: Posts Tagged: probability

UK National Lottery: now 21% more balls (rounded up)

This week, it was announced that from October the UK’s National Lottery, currently operated by Camelot and already providing a veritable Merlin’s cave of probability lessons for maths teachers, will be changing the rules for its main ‘Lotto’ draw. The main changes are that a new £1m prize will be added to the raffle element you didn’t know already happens, and that matching two balls will win a free ‘lucky dip’ ticket in the subsequent draw. The fixed £25 prize for matching three balls remains on the round table (even though it sometimes causes hilarious number gaffes).

But the Sword of Damocles hanging over Camelot’s changes is that there will be an extra ten balls to choose six from (59 instead of 49), dramatically lengthening the odds of winning all of the pre-existing prizes. This is our round-up of the media’s coverage of this mathematical “news”.

Probability of dealing four perfect hands of cards in a world of random shufflers

A couple of months ago (really? Two years?! Man!) I posted about an extraordinary coincidence: in a game of whist at a village hall in Kineton, Warwickshire, each of four players had been dealt an entire suit each. My post ‘Four perfect hands: An event never seen before (right?)‘ discussed this story. What really interested me was that the quoted mathematical analysis — and figure of 2,235,197,406,895,366,368,301,559,999 to 1 — appears to be correct; what lets down the piece is poor modelling. The probability calculated relies on the assumption that the deck is completely randomly ordered. Apart from the fact that new decks of cards come sorted into suits, whist is a game of collecting like cards together, so a coincidental ordering must be made more likely. Still unlikely enough to be worthy of mention in a local paper, maybe, but not “this is the first time this hand has ever been dealt in the history of the game”-unlikely.

Anyway, last week I was asked where the quoted figure 2,235,197,406,895,366,368,301,559,999 to 1 actually comes from. Here’s my shot at it.

The Chris Tarrant Problem

This is a puzzle I presented at the MathsJam conference. It’s a problem that gave me a headache for a week or so, and I thought others might enjoy it, too. I do know the answer, but I’m not going to give it away — you can tweet me @icecolbeveridge if you want to discuss your theories! (As Colin Wright says: don’t tell people the answer).

You’ve heard of the Monty Hall Problem, right?

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 $\gcd(a,b)=1$?

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.

According to the website, “all content will be free to access for the first two years of publication of the journal”. You can sign up for free email table of contents alerts.

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.