## "The odds of X are large": likely or unlikely?

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Murray Smith asks about a phrase in Joe Drape, "Looking for Zenyatta's Mr. Right", NYT 1/13/2011:

How Zenyatta will fare in her new career as a broodmare at Lane's End Farm is anyone's guess. She was a once-in-a-generation princess on the racetrack, winning 19 of 20 starts […]

Breeding, however, is more magic than math. […]

The Mosses and the people they have entrusted Zenyatta to know that **the odds of coming up with another horse like her are long**.

Murray observes that "The context makes clear that it is considered somewhat unlikely that Zenyatta will produce offspring of her calibre (the odds are against it), but the quote seems to say that it is likely (the odds of it are long)". Any idea what's happening? Any way to tell whether 'odds of' has always been ambiguous between "'for' and 'against'?"

In fact, "long odds" has always traditionally been used to mean "the odds placed on an outcome that has little or no chance of success". ("Short odds" is the opposite, though it is much rarer: 126 to 1 in the COCA corpus.) But confusion results when the traditional meaning of "long odds" is combined with the default association of *long* and *large*, along what the OED lists as sense **6.a.** of *odds*, "The chances or balance of probability in favour of something happening or being the case; probability, likelihood. Now usu. in (the) odds are: the likelihood is".

As a result, some people use large ("enormous", "huge", etc.) odds of X to mean that X is unlikely. Here are some examples from Google Books :

On the other hand, you might say that the odds of something happening are a million to one. Such odds might strike you as being so large as to rule out chance or coincidence. However, with over 6 billion people on earth, a million-to-one shot will occur frequently.

It would seem the odds of this [extraordinary coincidence] actually happening are impossibly large.

Since then, hunting for stratigraphic traps has been regarded as "elephant hunting" and the odds of finding them are equally large.

With different designs fitted on so many different makes over different years *t*he odds of having the one asked for are large indeed.

They understand that they won't always succeed, that the odds of overcoming transience are enormous, and that every institution they build is fragile.

The odds of winning are enormous, yet many of us dream of it. and inevitably someone will succeed.

But, can we expect children to have interest in life's real substance — in the arts, books, politics, philosophy, etc. — if parents have no interest? I think the odds of overcoming parental lack of interest are enormous.

I mean, as you well know, just playing in the big leagues, the odds of playing there and staying are so enormous.

The odds of all those things happening are huge, I'll admit, but not impossible.

Even if you don't believe in a higher power, the odds of you being conceived and born are huge, which makes you remarkable no matter what.

In fact, for those who think of odds in a mathematical sense, those examples are all backwards. Instead, large odds of X should mean that X is likely, as in these examples:

The fact that the odds of survival are greater for species better fitted to their environment gave rise to the well-known phrase "survival of the fittest."

Because surveys are endeavors that are ( 1 ) customized to each problem, and (2) constructed from thousands of detailed decisions, the odds of imperfections in survey statistics are indeed large.

For, if interpreters are confused about what it is they are interpreting, the odds of their producing incoherence are enormous.

With only a few seconds of drop time, no time for a backup parachute to open, and a real chance of slamming into the building or mountain from which you jumped, the odds of injury and fatality are enormous, far greater than with regular parachuting.

Death, injury, and/or impairment are consequences that occur very frequently to impaired drivers; the odds of these happening are enormous.

No matter how creative and careful a programming job you do, the odds of your being wrong are large, and in the health field, the odds are enormous.

*… *the odds of younger drivers experiencing accidents are twice as large as those for older drivers

On the other hand, where the odds of X are "tiny", "minuscule", or "small", the phrase always seems to mean "the odds in favor of X". I believe that this is due to the fact that "short odds" is hardly ever used, and so the confusion arises entirely at the large/long end of the scale.

This suggests that people are somewhat confused about what odds really are, or perhaps they're simply using the word in fixed expressions without thinking of the technical meaning even if they know it perfectly well. They're used to expressions like "a thousand to one shot" being used to mean that something is very unlikely to happen, and so they may think that if the odds of X are a thousand to one, then X is very unlikely to happen and has "large odds" — although technically, if they mean that the odds are a thousand to one against X, the odds in favor of X ought to be expressed as one in a thousand, which is a small number (relative to 1) rather than a large one.

It's possible that "the odds of X" is coming to be ambiguous between "the odds in favor of X" and "the odds against X". But I don't think so, because it this were happening, we'd expect to see evidence at the small end of the odds scale as well as at the large end.

Murray also points out that the "odds of" phrase seems to increased recently in frequency, as suggested by this Google n-grams plot

The COHA corpus supports the same conclusion:

Indeed, many of the pre-1970 instances of "the odds of" are in phrases like "what's the odds of two shillings more or less", "a man to whom even a first-class player can afford to give the Odds of a Rook and Knight has no claim to be ranked among Chess-players", "thou hast the odds of me", etc.

Perhaps this coincides with the spread of a more figurative use of *odds,* though it might alternatively reflect increased amounts of quasi-statistical language in the sources included in those collections. (I know that these explanations point in opposite directions — please feel free to investigate and to determine whether either of them is correct.)

[Update — a comment by Ian Preston is worth promoting:

I am not convinced that "odds" is a "a betting term originally" or just an extension of "betting talk". Looking at the history of its use in the OED, I get the impression that it began as a general term denoting unevenness which became attached some time in the 16th century to any inequality in the probabilities of an event occurring and not occurring and which was then picked up both in discussion of betting and of probability. The way the term is used has then diverged in those two contexts (presumably with influence in both directions) but it is not as if either set of users has any priority or is misapplying a term that originates with the other.

If you look at the early formal writings on probability in English then you find the use of the "odds *of* X" or "odds *that* X" as the preferred terms on one side and the "odds *against* X" on the other seems already well-established. (That looks to me to be the way terms are used by Abraham de Moivre in *The Doctrine of Chances* in 1718 and in his writings on annuities and by Thomas Simpson in *The Nature and Laws of Chance* in 1740, for example.)

This certainly accords with what I've been able to find out. But this leaves open the question of what has been happening in the past few decades, both to make "the odds of" (sometimes) ambiguous, and also to make it so much more common.]

## rootlesscosmo said,

January 16, 2011 @ 4:29 pm

"What's the odds?" to mean "what's the difference?" with the implication (from context) that the difference is negligible–i.e. a comment about (un)importance, not about likelihood–occurs in John dos Passos'

USA.## the other Mark P said,

January 16, 2011 @ 4:34 pm

I don't think "short odds" is the opposite of "long odds."

For me, "long odds" means something is very unlikely. "Short odds" is somewhere between likely and unlikely. "

Odds on" is when something is likely, and is therefore the opposite of "long odds".If the "odds are a thousand to one", then the event is most unlikely by definition. Only if the "odds

onare a thousand to one" is the event almost certainI accept that most people no longer know what "odds on" means, but I see no actual issue with the current usage of assuming "the odds" automatically implies an unlikely event – that is technically correct.

Worse, to my mind, is the use of "fraction" to always imply a small fraction (since mathematically 99 out of 100 is a fraction, just as 1 in a 100 is).

## empty said,

January 16, 2011 @ 4:56 pm

"Odds" is a betting term originally, isn't it? Bookies "give odds", meaning that based on an estimate of the chances they determine the payoff. Hundred to one odds means for each dollar you bet you get 100 dollars if your horse wins. This

roughlymeans that the bookie estimates that your horse has one chance in 100 of winning."Hundred to one odds" may mean the same thing as "one chance in a hundred" or "one percent chance" or "the probability is .01". Long (one doesn't say large) odds means low probability. But these idioms have unfortunately become mixed. Anyway, that's how I see it.

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January 16, 2011 @ 5:04 pm

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## Bloix said,

January 16, 2011 @ 5:14 pm

If probability of something is one in two (50%), then the odds of that thing are 1-to-1, or 1:1 (even odds). If the probability is one in four (25%), then the odds are 3-to-1. That's just the convention. So if the odds of something are 1000 to 1, the probability is 0.000999% – long odds, low probability. On the other hand, if the probability is 60%, then the odds are 2 to 3 – relatively short odds.

I would think that one reason long and short caught on for odds is precisely to avoid the confusion engendered by comparison with probability.

I'm just guessing here, but it may be that the use of "long" in connection with odds arose from the expression "long shot," meaning an arrow or rifle shot at a great distance that was therefore unlikely to hit the target, and by metaphorical extension first anything unlikely to happen and then any bet unlikely to pay off.

## empty said,

January 16, 2011 @ 5:16 pm

If the probability is one in four (25%), then the odds are 3-to-1.Yes, that's one reason why I said "roughly". The other is that the bookie is trying to make a profit.

## Mark F. said,

January 16, 2011 @ 5:46 pm

I'm pretty sure I've heard things like "The odds of that happening are 10-to-1 against", or "10-to-1 in favor". If you accept that you can drop the "in favor" or "against" when the context makes it clear, then you can harmonize both uses of "large odds".

## Fred said,

January 16, 2011 @ 6:17 pm

maybe the magnitude value ("large") is either referring to the denominator in the proportion ("1 in a million") or the percentage estimate of the probability ("99 %).

## Ian Preston said,

January 16, 2011 @ 7:30 pm

I'd find it more natural to say that the odds are "high" than that they are "large" and that does seem to be substantially more common. It also seems less ambiguously used. I couldn't find any examples of "high" odds meaning low probability though if you stretch the vertical dimension hyperbolically to talk about "stratospheric" odds then you can find examples of the meaning switching. "Tall" odds, however, seems, like "long" odds, always to denote low probability.

## linda seebach said,

January 16, 2011 @ 7:33 pm

When I was working at a newspaper, I had on occasion to explain to colleagues how odds and probabilty are related. At least one was distressed that odds didn't add up to one.

## Rubrick said,

January 16, 2011 @ 8:12 pm

It's unsurprising that people get confused about this, since to most the derivation of "odds" is opaque. People know it refers to a ratio, but since one hears both "3-to-1 chance" and "1-in-4 chance", it's not at all obvious whether "large", "small", "long", etc. refer to the numerator or the denominator.

Unfortunately, developing an intuitive grasp of this is difficult, since the science of probability has a fairly steep, or else shallow, learning curve.

## Mike Anderson said,

January 16, 2011 @ 8:18 pm

Rubrick:

probability = wins / (wins + losses)

odds = wins / losses

I plump for the shallow (short?) learning curve.

However, intuition is no more useful in guessing the solution to a probability problem than it is in trying to select an anniversary gift for your wife. One must learn the rules and follow them scrupulously.

## John said,

January 16, 2011 @ 8:59 pm

"Large odds"? Strikes me as the kind of thing someone who doesn't play the odds too often would say.

## David L said,

January 16, 2011 @ 9:03 pm

I seem to recall — I had an uncle who liked the ponies — that when the announcers of a race were giving odds, the default assumption was that the odds were quoted against the outcome. In other words, 5-to-1 meant not very likely, 100-to-1 meant very unlikely, so that long odds indeed meant low probability of success. But when the odds were short, they would say 2-to-1

on— meaning a 2 out of 3 expectation of success — hence 'odds-on favorite' for a horse expected to win.## David L said,

January 16, 2011 @ 9:08 pm

And I now see that the other Mark P already made the same point. What are the odds of that?

## Ian Preston said,

January 16, 2011 @ 11:00 pm

To say that there is a "narrow chance" of something happening is to say that it is unlikely. To say that the odds of it happening are "narrow" is, on the contrary, to say that it is likely (and to say that its odds are narrowing is to say that it is becoming more likely).

Some examples:

This strikes me as particularly confusing. Unless I am myself confused, then narrow odds are short odds not long odds but it is not surprising to find people talking, say, of "triumphing over narrow odds" or "winning despite narrow odds" or the like.

## Garrett Wollman said,

January 16, 2011 @ 11:41 pm

I always try to avoid using odds, for precisely the reason that I can never be sure of being understood (or even of having understood it myself). Much simpler to stick with probabilities.

I remember reading a rant, not sure whether it was here or on Cosma Shalizi's blog, about the misleading use of odds ratios — that is, the ratio of the odds of one outcome to the odds of another outcome — in reporting medical research.

## Gary M said,

January 17, 2011 @ 3:47 am

Many math texts distinguish between the "odds" of you winning, say 1:99 for an event with 0.01 probability, and the "house odds" of the house winning, which are the reverse, or 99:1. House odds are what you hear quoted by betting establishments. Since this is where most people learn the concept of "odds", house odds are what they think of first, not their own odds.

In popular language, both are simply called "odds" so it's no wonder there's confusion. Large house odds mean an event is very unlikely ("the odds are long"), while large odds mean an event is very likely.

## D said,

January 17, 2011 @ 5:40 am

However, when gambling, odds are often (almost always?) represented as the multiplier on your investment. So if team A is very likely to win, their odds might be 1.1, while team B's are 5. I.e, less likely = larger.

## Spell Me Jeff said,

January 17, 2011 @ 9:53 am

MarkF is onto something. The preposition tells all. These would seem to mean the same thing:

[with respect to X happening] the odds are long/hugeand

The odds against X happening are long/hugethe second of which seems unambiguous (to me, anyway)

The idea gets confused when we swap in "of," as in

The odds of X happening are long/hugeA less confusing construction with the same intent would be:

The likelihood of X happening is hugewhich also strikes me as unambiguous.

## Ellen K. said,

January 17, 2011 @ 10:11 am

I'm surprised by the idea that long odds = large/huge odds. In most of the constructions with odds, I "large" doesn't even make sense.

In the initial quote, I had no trouble understanding "long" as meaning it's not likely. If I substitute "large", it means the opposite. I don't see any ambiguity. It's simply that "long" and "large" aren't equivalent. And that may have to do with different meaning of the word "odds", different realm, that each of these words belongs to. Betting odds can't be large or small, seems to me. And where "odds" can be interchanged with "likelihood", "long" can't be used.

## Jonathan Mayhew said,

January 17, 2011 @ 12:04 pm

I think we've found a case where minuscule and astronomical can mean the same thing: "He's facing astronomical odds." Or "The odds of his success are minuscule."

## Chandra said,

January 17, 2011 @ 12:24 pm

There's something very quantum-leapish about this whole discussion.

## GeorgeW said,

January 17, 2011 @ 3:00 pm

Odds can also be good or not good.

Q. What are the odds she'll make it through the operation?

A. Very good.

Q. What are the odds of winning the lottery?

A. Not good

## Paul Kay said,

January 17, 2011 @ 3:39 pm

Mark F. and Spell me Jeff seem to have said what I would have said (and what I think is also clear in Mark's original presentation). If one speaks of "odds for" or "odds against" an event's occurring, there's no possibility of misunderstanding. If one speaks of "odds of" or "odds that" one invites one's interlocutor into an interpretive swamp. (Which is not to say it's not a frequently tendered invitation.) Several commenters have also mentioned "odds on" as indicating short odds, in fact odds better than even. If a horse is an "odds on favorite" that means a winning $2.00 ticket pays the bettor something between $2.00 and $4.00 at collection; that is the bettor is "giving" the odds (to the track) rather than "taking" them. I've heard that, too, but I think of it as a separate usage.

## Mr Fnortner said,

January 17, 2011 @ 3:44 pm

Can someone here make logical sense out of expressions of the sort,

5 times smaller,as in "X is 5 times smaller than y," when what is meant is1/5 as large?## Murray Smith said,

January 17, 2011 @ 4:02 pm

@ Paul Kay: Your reading (and Mark F. and Spell me Jeff's) are just what I would have said. But as the Google ngram data show, that leaves lots of folks in the swamp. I don't doubt that context provides the path out of the swamp almost every time, but it's curious that usage doesn't settle into a more reasonable pattern. Yet.

## Peter said,

January 17, 2011 @ 5:12 pm

The other Mark P said:

> Worse, to my mind, is the use of "fraction" to always imply a small fraction (since mathematically 99 out of 100 is a fraction, just as 1 in a 100 is).

Is this really all that bad? There are a number of other similar usages: for instance, the use of “a number” to mean “a reasonably large number” — mathematically, 0, 1, 1/2, and –17 are perfectly good numbers too, not to mention √2, (1–2

i), or אε₀. Both this, and ‘fraction’, seem perfectly reasonable cases of meaning being context-dependent; unlike the “long odds” vs. “large odds” example, it doesn’t seem like they’d cause confusion very often.## Ken Brown said,

January 17, 2011 @ 6:27 pm

David L has it right. Odds are "long", "short", or "even". They can be "on", by default they are assumed not to be. They can be "against". But not "large".

It is betting talk. Hundreds of thousands of people use it and understand it every day. Loads of people in the pub I am posting this from do. Some of them left school at 16 and work on building sites or driving vans.

But it is tecnical language. People who don't bet often get it wrong. Just as people who don't do linguistics or genetics or art history or philosophy get the jargons of those fields wrong.

## Coby Lubliner said,

January 17, 2011 @ 8:35 pm

Mark P and Peter: "fraction" and "number" are only two among many words whose meaning (or use) in everyday language differs from their meaning in mathematics: factor, function, product, set, union, intersection, field, ring, parameter…

As regards

odds, it seems to me that, sinceoddmeans (more or less) 'unlikely' ("something odd happened to me"), the underlying meaning ofoddsis 'unlikeliness' (MWCD11 gives 'degree of unlikeness' [whatever that means] as an obsolete meaning).## Joshua said,

January 17, 2011 @ 8:50 pm

There's a song by Amy Grant and Vince Gill called "House of Love" which seems to me to include an illustration of this problem in the lyrics: "But when something this strong/Gets a hold on you/The odds are 99 to 1/It's got a hold on him too." From the context, I think the song is trying to say something is likely, but "99 to 1" odds make it sound very unlikely.

## a George said,

January 18, 2011 @ 3:33 am

ah, the last word! "Odds" is the market value of a probability. This is merely being more specific than empty, above. I am not a betting man, and when I have seen "long" used it has been in the expression "a long shot", which to me signifies "little probability of occurring".

## the other Mark P said,

January 18, 2011 @ 5:00 am

Mark P and Peter: "fraction" and "number" are only two among many words whose meaning (or use) in everyday language differs from their meaning in mathematics: factor, function, product, set, union, intersection, field, ring, parameter…In general the use of "number" in normal English is pretty much identical to the mathematical concept "whole number". If I say "think of a number", many people will pick 1 or 2, and few will pick much more than about 30. I think most people accept that 1 and 2 are numbers, just as 1,534,870 is. Yet, as Peter says, in one particular circumstance we are expected to understand that a

largenumber is meant.If I ask a person in everyday life to give a fraction, they will give a fraction between 0 and 1. What would mathematically be called a "proper" fraction. Almost everybody would be quite happy that two-thirds is a fraction in the standard English meaning of the word. Yet if I say that he only earned a fraction of his brother, it is implied that a

smallfraction is meant.In both cases there are times when we are expected to understand that the terms differ from the common English meaning, as well as the more technical mathematical meaning.

## Ian Preston said,

January 18, 2011 @ 7:59 am

I am not convinced that "odds" is a "a betting term originally" or just an extension of "betting talk". Looking at the history of its use in the OED, I get the impression that it began as a general term denoting unevenness which became attached some time in the 16th century to any inequality in the probabilities of an event occurring and not occurring and which was then picked up both in discussion of betting and of probability. The way the term is used has then diverged in those two contexts (presumably with influence in both directions) but it is not as if either set of users has any priority or is misapplying a term that originates with the other.

If you look at the early formal writings on probability in English then you find the use of the "odds

ofX" or "oddsthatX" as the preferred terms on one side and the "oddsagainstX" on the other seems already well-established. (That looks to me to be the way terms are used by Abraham de Moivre inThe Doctrine of Chancesin 1718 and in his writings on annuities and by Thomas Simpson inThe Nature and Laws of Chancein 1740, for example.)## The Blackboard » Odds are: Language Log Reveals all! said,

January 18, 2011 @ 1:08 pm

[…] my surprise, on January 16, Mark Liberman of Language Log posted "The odds of X are large": likely or unlikely?, explaining why people often become confused when discussions involve phrases using the term […]

## slobone said,

January 18, 2011 @ 3:40 pm

At the racetrack, the odds quoted are always "against", never "for". Therefore the ambiguity of "odds of" can only be resolved by changing the "of" to "against," since the story is about racing.

I was once foolish enough to bet against a horse being ridden (or whatever you call it in a trotting race) by a visiting woman jockey from the USSR. Everybody but me seemed to know that, of course, as a courtesy to a guest, the other jocks would let her win. The odds dropped to 1/2 to 1, and sure enough, she won.

## slobone said,

January 18, 2011 @ 3:42 pm

PS by the same token, if the outcome really was a "mathematical" certainty, then rightly the odds should be 0 to 1. I'm pretty sure that's never happened…

## iching said,

January 18, 2011 @ 7:20 pm

My feeling is that the confusion arises because the technical term

oddsin mathematical statistics is also used in everyday language, sometimes with a different meaning.There is a formula that converts

oddstoprobability(also known aslikelihood), namelyo = p/(1-p)forodds fororodds onan event (used when p>=0.5) oro = (1-p)/pforodds againstan event (used when p<=0.5). While probability must lie between 0 and 1, odds (when used in this way, specified as eitheronoragainst, the default beingagainst) must vary from 1 to infinity. Odds can be short, narrow or small (i.e. close to 1) or long, wide or large (or astronomical!)However, the online OED confirms that <i, in distinction to the technical definition,

oddscan simply meanprobabilityorlikelihood:"The chances or balance of probability in favour of something happening or being the case; probability, likelihood. Now usu. in (the) odds are: the likelihood is. it is odds (that, †but): it is probable; the likelihood is (now rare)."Hence, in a phrase like "the odds of a large asteroid hitting the Earth are large/small" depends on whether

oddsis used in the technical/lay sense. In the former, "odds of" must be interpreted as meaning "odds against", in the latter "odds of" simply means "probablity of", "likelihood of".## Ray Dillinger said,

January 19, 2011 @ 10:36 pm

I have "translated" bettor-to-mathematics before, and Ken Brown and David L have it exactly right. As used by bettors, gamblers, and those who analyze betting games, these are technical terms with a very precise meaning.

One that I hadn't heard before I worked from that book (Doctor Z's Beat The Racetrack) was "oddment", which I still haven't seen in that sense anywhere else. An "oddment" when talking about gambling games is an equal proportion of probability associated with a particular outcome in a proposition having several possible outcomes. For example, in the aforementioned horse-race where your best handicapping efforts say a horse has 5-to-2 odds of winning and the bookie is offering 2-to-1, your analysis shows 7 oddments 2 of which are favorable to a win bet, and the bookie's shows 3 oddments 1 of which is favorable to a win bet.

A quick calculation to see if the odds favor betting is to multiply opposite-outcome oddments from your odds and the bookie's. In this case because 5×1 is more than 2×2, you shouldn't make the bookie's bet.

## Paul Kay said,

January 20, 2011 @ 1:24 pm

Mr Fnortner asks, "Can someone here make logical sense out of expressions of the sort, 5 times smaller, as in "X is 5 times smaller than y," when what is meant is 1/5 as large?" Indeed, the patent illogic of this form of speech is proven by the fact that it is absolutely standard at every level of formality in French. For example "fois plus petit que" 'times smaller than' (immediately preceded by a number word in the first twenty instances) gets 1,750,000 raw Google hits. Expressions of this form are common in popular French speech, in scientific writing, and in classic authors like Balzac. The fact that this form is present in all levels of French and in many varieties of non-prestige English — not to mention the fact the Mr Fnortner himself has no trouble interpreting it — shows that logic has nothing to do with the matter. There's no

logicalreason "X is 5 times smaller than Y" can't mean 'If X is of size A, Y is of size A/5.' It's a convention they teach you in school, like that black and white aren't colors — which of course they have to teach you only because you already know they are. As a faithful reader of Language Log, Mr Fnortner should know that attributing lack of logic to forms of speech other than one's own is a risky business.## The Ridger said,

January 20, 2011 @ 3:28 pm

@Paul Kay – it's also absolutely standard in Russian – also things like "twice as short" meaning "half as tall", or "will make three times less profit than last year" for "one third the profit". It's amazing (or maybe not) how problematic these are for learners.

## The Ridger said,

January 20, 2011 @ 3:32 pm

@slobone: try "driven" and "driver" for "ridden" and "jockey". :-)

## Paul Kay said,

January 20, 2011 @ 5:18 pm

@The Ridger. Yes, and, e.g., Spanish "veces menos que" 'times smaller than' yields over two million Google hits. I'm neither a great polyglot nor a terribly energetic scholar, but I'd bet more Indo-European languages have this construction than don't. I'm afraid we're drifting off topic, though.

## Paul Kay said,

January 20, 2011 @ 7:22 pm

@myself. I wrote, "There's no logical reason "X is 5 times smaller than Y" can't mean 'If X is of size A, Y is of size A/5.'" Backwards! Should be: There's no logical reason "X is 5 times smaller than Y" can't mean 'If Y is of size A, X is of size A/5.'

## Maureen said,

January 20, 2011 @ 9:53 pm

1/2 – 1 isn't really low. If the odds of the Russian lady winning had been

1-100, then you would really have known you were skunked. ;)

## “The odds of X are large”: likely or unlikely? | tonycreary said,

January 23, 2011 @ 7:24 am

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