What is probability?

I haven't seen any particularly interesting (or unusually egregious) uses of statistics in the last few days, so it's time to have a more philosophical outing.  Of course, if you see anything in the news you think is worthy of a post, email me (evans [at] stats [dot] ox [dot] ac [dot] uk).

So to the title of the post - what is probability?  Probability can be viewed as just a mathematical construct, which we'll explore some other time, but even with the mathematical rules in place, there are various interpretations of what a probability actually is.  At school we usually think about rolling dice and flipping coins, so let's start there.  I flip a 50p coin: what's the probability that it comes up heads?  You'd probably [sorry] say it's 'evens', 'a half', 'fifty-fifty', or (as mathematicians generally prefer), '0.5'.

Fast food, dodgy infographics, implausible claims

The Daily Mail reported on Wednesday that US restaurant portion sizes had quadrupled(!) since the 1950s, a claim faithfully reproduced from the Center for Disease Control's (CDC) website.  The article comes complete with this infographic.

Now, it should be immediately obvious (if you remember your GCSE maths) that something is not quite right with this picture: the fountain soda looks truly enormous.  Taking the numbers in the diagram on faith for the moment (but see below), the volume of a cup has increased from 7 oz (200 ml) to 42 oz (1.25 l), a worrying factor of six increase.  In accordance with this, the 'NOW' cup on the diagram is 6 times taller than the 1950s cup.

Non-existent recidivism statistics wrongly reported

This article from the BBC News website caught my eye today, headlined "Reoffending rates reach record level".  It also says that 

Ministry of Justice officials say the figures show a "clear trend" of a rising re-offending rate.

The Independent makes some similar claims.  The phrase "re-offending rate" is simple enough: it's the proportion of people who, having been convicted of a crime, go on to commit another one within a year, or within five years, or whatever.  Right?