What are "the odds"?

Andrew West linked on Twitter yesterday to this story:

Take That: Christmas odds slashed after X Factor The odds of Take That album Progress being Christmas number one have dropped to 2/5 after their first performance as a five piece in 15 years on X Factor.

He said the use of ‘slashed’ in the headline was confusing, and I agreed – even accounting for the bizarre form of headline-grammar where nouns act as adjectives until you get nonsense like “Slough sausage choke baby death woman jailed”.

But following a short but helpful discussion and having read up on things, I think the issue is not so much to do with ‘slashing’ but to do with ‘odds’ being a rather confusing concept. According to the OED, “the odds” refers to the relative stakes offered by the two parties (gambler and bookmaker). It is a property of a bet, not of an event. Theoretically, it ought to be given by

odds = (1 ÷ probability) -- 1

but in practice this is always adjusted to add a profit margin for the bookmakers. If the bookmakers are doing their jobs, then a likely event has short odds and an unlikely event (or ‘long shot’) has long odds. To change long odds into short odds, you cut (or slash) them. The bookmakers have not changed the probability of something happening, they’ve changed the terms on which they’ll accept the bet.

All fair enough, but I don’t use the word ‘odds’ that way and I bet you don’t either. I use ‘odds’ to mean ‘probability’ – as in, “what are the odds of that happening?”. This is a colloquial usage, and it isn’t really defined well mathematically. However, my instinct would be to define it as

odds = probability

and indeed, Collins Dictionary defines “odds” as “the likelihood that a certain state of affairs will be so”, and some news websites use it this way.

The BBC article in question discusses ‘odds’ as a property of an event, not a bet. It refers to “the odds of Take That… being Christmas number one”, not “the odds offered on Take That… being Christmas number one”, and no bookmaker or bet is mentioned in the next three paragraphs. This, I think, is as it should be. That a band is likely to be Christmas number one with its original line-up not seen for fifteen years is perhaps newsworthy, but that Ladbrokes have therefore increased the price of betting that this will happen surely isn’t (although the BBC does like to report people’s reactions to news instead of the news itself).

But if they’re discussing the probability of Take That being number one, then 2/5 isn’t a useful figure. First, it includes a profit margin; second, William Hill disagree (or have narrower margins) and say 4/9; third, it’s almost totally opaque to non-gamblers, who are liable to assume Take That have a 40% chance of being Christmas number one; and fourth, to come back to the original issue, ‘slashed’ in a newspaper headline (almost) always refers to a decrease, and while the bookmakers’ odds have been cut from 28/1 to 2/5, the event’s odds have arguably increased from around 34% to around 71%. In this sense, ‘odds’ is one of those unusual words that are their own antonyms. The BBC’s stylebook can say what it likes about the difference between ‘odds’ and ‘probability’ but their headlines should be immediately understandable to people who haven’t read it, and I don’t think this one was.

I for one think the best policy would be not to use ‘odds’ in a headline at all unless you mention a bookmaker. In this case, I’d write the headline as “X Factor boosts Take That’s chances of Christmas number one”, or even change the focus of the headline to “Take That favourites for Christmas number one since X Factor”.

Or better still, I’d simply refuse to report this irrelevant drivel at all.