Can odds be awkward? I’d put money on that...

Assiduous readers of understandinguncertainty.org will know that we often refer to odds. Pretty well everyone will have heard of odds, and will at least know that they have something to do with how likely something is to happen. But beyond that, it can get trickier, as an entry in a blog about language has reminded us.

Mathematically, it’s all pretty straightforward. If something has a probability $p$ of happening, then the odds of it happening are

So, since it’s tipping with rain now, my probability of getting wet if I go out to the shops in an hour might be [math]p=0.9[/math], and that means the odds of getting wet are [display]\frac{0.9}{(1-0.9)},[/display] which is [math]\frac{0.9}{0.1}[/math] or 9. On the other hand, I looked at the weather forecast and it said it might clear up later, so if I wait till this afternoon to go shopping, maybe it will be as likely to rain as not. So my probability of getting wet would be 0.5, and the odds of getting wet would be [display]\frac{0.5}{(1-0.5)},[/display] which is [math]\frac{0.5}{0.5}[/math] or 1. Rain is less likely later on, so the probability is less and the odds are less. Generally, the bigger the probability, the bigger the odds.

It’s not quite that simple, though. Statisticians use odds rather than probability, sometimes, because odds can make things easier to describe and work with. But there’s more to it than that. Statisticians didn’t invent odds; the language originally came from betting and gambling. And talking about odds isn’t confined to statisticians and bookmakers. Anyone might talk of a very surprising event and say something like “The odds of that are a thousand to one”. Hang on... a thousand looks like a big number, we’re talking about a surprising event, that is, one with low probability. But I just said that big odds go with big probability. What’s going on?

The complication is that, in a betting context, there are two different kinds of odds for any given event. There are the odds on the event, which (in terms of probability) are what I described earlier. If the probability is [math]p[/math], the odds on the event are [display]\frac{p}{1-p}.[/display] But there are also the odds against the event happening. For an event with probability [math]p[/math], the odds against are [display]\frac{1-p}{p}.[/display] The fraction is the other way up.

So if I go shopping in an hour (probability of getting wet = 0.9), the odds on getting wet are 9 (or, in betting language, “9 to 1 on”), but the odds against getting wet are [display]\frac{1-0.9}{0.9}[/display] which comes to 1/9. If I leave my shopping till tomorrow, when the forecast looks a lot better and I’d put the probability of rain at [math]p=0.1[/math], the odds on getting wet are [display]\frac{0.1}{1-0.1}[/display] which is 1/9, and the odds against getting wet are [display]\frac{1-0.1}{0.1},[/display] which comes to 9 (or in betting language “9 to 1 against”).

Traditional betting language in the UK (and in many other countries) would sometimes use odds on an event, sometimes odds against. Which is which would always be clear, at least to those familiar with betting jargon; there are conventions about whether to quote odds on or odds against in different situations, and about the way they are stated. But where this language has carried over into everyday speech, things aren’t always very precise. People might say that the odds of some event are large when it’s very likely (large odds on), or when it’s very unlikely (large odds against), and it isn’t always clear which they mean.

This is where the language blog comes in. The author of the post, Mark Liberman is a professor in linguistics and in computer science at the University of Pennsylvania. His post uses data from various sources to try to track what’s going on in the language people actually use. (He uses Google’s “n-grams”, which track language in digitised books, and two so-called “corpora”, or databases of language use, COCA which covers contemporary American English, and COHA for historical American English.) He finds a major rise since the 1980s of use of the phrase “odds of”, though it’s not clear whether this has led to greater confusion between odds on and odds against.
The message in terms of understanding uncertainty must surely be that, if you read something about odds, you should always work out carefully what kind of odds are meant - if indeed it is possible to figure it out!