This picture is cropped from the wonderful item #50D1-5830 by Hideta Nagai, whose View Finder Photography website has some spectacular shots.
Monday, 14 March 2011
Sunday, 13 March 2011
I've only recently started this intermittent blog, and a few days ago I sent off an email to a friend, notifying him of its existence. We haven't communicated for a few months, so it was amusing to get back the response:
It might seem like the question is: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages?" Pretty small, I suspect. But actually, the question is really: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages, and given that I've just received an email from you just as I'm writing you one?". That is, what is p(x|x) (the probability of x having happened, given that x has happened)? Well, it's one. You can't get less unlikely than that!
Okay, that seems a little unsatisfactory. It still seems somehow to be remarkably unlikely. What's the probability it will happen again? Very small. But what's the probability that some weird coincidence will happen again? Now that is rather high.
Let's assume that we think some event has a probability of one in a gazillion of happening (the probability prior to it actually having happened, that is). But there are equally gazillions of unlikely things that could happen. Say you get an email from a friend just as you were thinking of them. But they might have phoned you, or texted you, or visited you, or written to you. Or you might have seen them on TV, or read about them, or about someone with the same name. And you could have been thinking of any of your friends, or of anyone else, or of anything else.
There are oodles of possible unlikely coincidences. What are the odds that one of them happens?
The way probability works, it's easier to calculate the chance of none of them happening. Let's say the odds of each one of these things happening is one in N, where N is very large (one in a billion, one in a trillion, or more). So the probability is 1/N, and the probability of it not happening is 1-1/N (very nearly, but not quite, certain that it won't happen).
Now let's say the number of unlikely things that might happen is also this huge number N (it could be 10N, or N/10; the calculation is cleaner using N, but the overall flavour of the result still holds for other values).What is the probability that none of the N unlikely things happens? That is, what is the probability that the first thing doesn't happen, and the second thing doesn't happen, and ... all the way up to and the Nth thing doesn't happen?
We just multiply the individual probabilities together, so we get (1-1/N)^N. That's the probability of no coincidences, so the probability of at least one coincidence is p = 1-(1-1/N)^N. For N larger that about 100, p is about 63% (for 10N it's 99.99%, for N/10 it's 10%). That's a pretty good chance of a weird coincidence. And that's just today!
The moral is: when individual events are unlikely, but there are also a lot of events that could happen, something will almost certainly occur.
So, that email: unlikely coincidence? That coincidence, yes; some sort of coincidence happening, not really.
(interesting stuff about stuff ....)Now, I know he doesn't really think it's telepathy, but some people do get exercised about this sort of coincidence. But things like this happen all the time. And it's easy to see why.
Hey, I was in the middle of this when YOUR e-mail came in! Telepathy?
It might seem like the question is: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages?" Pretty small, I suspect. But actually, the question is really: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages, and given that I've just received an email from you just as I'm writing you one?". That is, what is p(x|x) (the probability of x having happened, given that x has happened)? Well, it's one. You can't get less unlikely than that!
Okay, that seems a little unsatisfactory. It still seems somehow to be remarkably unlikely. What's the probability it will happen again? Very small. But what's the probability that some weird coincidence will happen again? Now that is rather high.
Let's assume that we think some event has a probability of one in a gazillion of happening (the probability prior to it actually having happened, that is). But there are equally gazillions of unlikely things that could happen. Say you get an email from a friend just as you were thinking of them. But they might have phoned you, or texted you, or visited you, or written to you. Or you might have seen them on TV, or read about them, or about someone with the same name. And you could have been thinking of any of your friends, or of anyone else, or of anything else.
There are oodles of possible unlikely coincidences. What are the odds that one of them happens?
The way probability works, it's easier to calculate the chance of none of them happening. Let's say the odds of each one of these things happening is one in N, where N is very large (one in a billion, one in a trillion, or more). So the probability is 1/N, and the probability of it not happening is 1-1/N (very nearly, but not quite, certain that it won't happen).
Now let's say the number of unlikely things that might happen is also this huge number N (it could be 10N, or N/10; the calculation is cleaner using N, but the overall flavour of the result still holds for other values).What is the probability that none of the N unlikely things happens? That is, what is the probability that the first thing doesn't happen, and the second thing doesn't happen, and ... all the way up to and the Nth thing doesn't happen?
We just multiply the individual probabilities together, so we get (1-1/N)^N. That's the probability of no coincidences, so the probability of at least one coincidence is p = 1-(1-1/N)^N. For N larger that about 100, p is about 63% (for 10N it's 99.99%, for N/10 it's 10%). That's a pretty good chance of a weird coincidence. And that's just today!
The moral is: when individual events are unlikely, but there are also a lot of events that could happen, something will almost certainly occur.
So, that email: unlikely coincidence? That coincidence, yes; some sort of coincidence happening, not really.
Sunday, 6 March 2011
I write many documents in LaTeX, and it is great for things like bibliographies, cross-referencing, and defining styles. But the default styles, such as 'article', are a bit wasteful of space. For example
This is particularly a problem if you are preparing a document with strict page limits, such as a research proposal or a conference paper. Those places often put their own style requirements. Some provide a LaTeX style file, which is fine (provide the style actually works!), but others are a little less helpful.
Here is a typical requirement:
Not only is it conformant, it also has more text on it than the plain `article' version. But we can do better. One reason the article style is so fascistic about its left and right margins (try to get more text by dropping a point size, and it reduces the line length!) is that very long lines are hard to read: you lose your place as you track back at the end of a line. A "11pt text, 2cm margin" format results in lines too long to read comfortably. So let's do what the newspapers do, and go to multi (here, two) columns. This actually saves space, as short section titles no longer take up a whole full width line.
Also, let's take the title and abstract into the main body, and tighten up the space around section headings using the 'titlesec.sty' package. Finally, use some 'negative leading' to get the lines a little closer together (without looking cramped), and we get:
The 'titlesec' package also allows some fancier formatting of section headers (for example, when things get a little squashed, I like to put a rule over major section titles, to aid the eye), but that's another story.
gives the page shown on the right.\documentclass[a4paper]{article}
\title{Your Title Here}
\date{}
\begin{document}
\maketitle
\begin{abstract}
Lorem ipsum ...
\end{abstract}
\section*{Introduction}
...
\end{document}
This is particularly a problem if you are preparing a document with strict page limits, such as a research proposal or a conference paper. Those places often put their own style requirements. Some provide a LaTeX style file, which is fine (provide the style actually works!), but others are a little less helpful.
Here is a typical requirement:
Font size 11 is the minimum font that is acceptable, and the minimum margin in all directions is 2cm. For accessibility purposes, a sans-serif font style such as Arial or Helvetica should be used as these are more easily readable to those with visual impairment. For the same reason, type should be justified only on the left hand side.How do we get LaTeX to meet these requirements? Well, there are a few useful packages. 'geometry.sty' lets the margins be easily set. So we can get a conformant document with:
\documentclass[a4paper,11pt]{article} % 11pt font
\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry} %min margins
\renewcommand{\sfdefault}{cmss} % sans serif
\renewcommand{\familydefault}{\sfdefault} % for whole doc
\raggedright % left justified
\parindent 1em % because \raggedright stops indenting
\begin{document}
...as above
\end{document}
Not only is it conformant, it also has more text on it than the plain `article' version. But we can do better. One reason the article style is so fascistic about its left and right margins (try to get more text by dropping a point size, and it reduces the line length!) is that very long lines are hard to read: you lose your place as you track back at the end of a line. A "11pt text, 2cm margin" format results in lines too long to read comfortably. So let's do what the newspapers do, and go to multi (here, two) columns. This actually saves space, as short section titles no longer take up a whole full width line.
Also, let's take the title and abstract into the main body, and tighten up the space around section headings using the 'titlesec.sty' package. Finally, use some 'negative leading' to get the lines a little closer together (without looking cramped), and we get:
\documentclass[a4paper,twocolumn,11pt]{article}The result has about twice as much text as the original 'article' style, and is fully conformant to the stated requirements.
\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}\renewcommand{\sfdefault}{cmss}
\renewcommand{\familydefault}{\sfdefault}
\raggedright
\parindent 1em
\usepackage[small,compact]{titlesec} % section headings in smaller font size, less whitespace
\renewcommand{\baselinestretch}{0.9} % closer lines
\begin{document}
\section*{\Large Your Title Here}
\section*{Executive summary}
Lorem ipsum ...
\section*{Introduction}
...
\end{document}
The 'titlesec' package also allows some fancier formatting of section headers (for example, when things get a little squashed, I like to put a rule over major section titles, to aid the eye), but that's another story.
Thursday, 17 February 2011
I got my electricity bill last week, and I noticed a nifty new feature that's been added to it: a predicted electricity cost for the next year. And very interesting that prediction is, too.
Here is my annual expenditure on electricity for the last few years (the numbers are ridiculously low for several related reasons that I won't go into here):
2003: £45.36
2004: £33.14
2005: £38.26
2006: £51.04
2007: £55.71
2008: £48.74
2009: £48.30
2010: £56.16
The graph shows a noisy but nevertheless upward trend, consistent with inflation more than with any increased usage on my part.
What do you think my electricity company's prediction for my 2011 bill is? A prediction that I am assured is based on "actual readings", according to my account. £60? Maybe as much £70? No. It is £488.77.
So, what does npower know about short term electricity price inflation that I don't?
Here is my annual expenditure on electricity for the last few years (the numbers are ridiculously low for several related reasons that I won't go into here):
2004: £33.14
2005: £38.26
2006: £51.04
2007: £55.71
2008: £48.74
2009: £48.30
2010: £56.16
The graph shows a noisy but nevertheless upward trend, consistent with inflation more than with any increased usage on my part.
What do you think my electricity company's prediction for my 2011 bill is? A prediction that I am assured is based on "actual readings", according to my account. £60? Maybe as much £70? No. It is £488.77.
So, what does npower know about short term electricity price inflation that I don't?
Tuesday, 15 February 2011
Without conscious decision, the calculation flashed through my head as follows: "Eight for a pound; that's half a crown each", then a slight pause before the next step: "so that's 12-and-a-half pence each."
40 years: oh dear, and I still think in old money! (Not all the time, I hasten to add. It's just that "eight half crowns to the pound" was so well drummed in that it's still the natural association. Then there's always 6/8d.) But, to the day: wow.
Never mind that. What about half a crown for a single pear! Outrageous! (OTOH, 12.5p is really cheap...)
Sunday, 13 February 2011
Now, I know King's Cross quite well, I thought. It has eight platforms on the main concourse, and also platforms 9 to 11 hidden away in their own private little siding. (Forget finding platform 9-and-3/4s: I've know people fail to find the real but well-hidden platform nine!) But platform 0? Srsly? Only computer scientists start counting from 0!
So I was disappointed to discover that once the refurbishment is completed, the platforms will be renumbered, starting from the much more unimaginative 1. (Presumably resulting in hilarity for all concerned as the zombified commuters, with the current numbering scheme hardwired in their cortexes, start making off-by-one errors, and end up scattered all over the country.)
Subscribe to:
Posts (Atom)
