# “One in a million” or “A million to one”?

Last week I took part in an English course, and the teacher was constantly saying a million to one (when he meant "an extremely small possibility").

Is this correct? Is it the same as one in a million?

In polish we say the equivalent of one in a million (pl: jedna na milion) — which, by the way, seems much more correct (considering the mathematical sense).

Well, the way odds are described in English,

• Three to one means one in four (a 25% chance)
• Two to one means one in three (a 33.3% chance)
• Evens (in other words, one to one, although the latter is not something I have ever seen used) means one in two (a 50% chance)

So strictly speaking, a million to one would mean one in a million-plus-one, which is a fractionally smaller probability than one in a million. However, I strongly suspect that the popular usage of these terms is not specifying absolute odds but simply describing something very unlikely. In that case, the two are effectively equivalent.

However, when used in this casual, hyperbolic manner (as you say, just to mean "a very small possiblity"), to one has a distinct extra meaning compared to in one; it emphasises the adversity faced by the person (or thing) facing these odds. "A million to one" implies that victory would require a struggle against overwhelming difficulties, because the wording places the odds of failure in opposition to the chances of success. "One in a million" can simply mean a lucky accident. So if you are describing an overweight, unfit person's chances of winning the London marathon, then it is better to say the odds against it are "a million to one". But if your story is of somebody who picked up a stone in a riverbed and found a lump of gold underneath it, that could be called a "one in a million" lucky break. Both could be used in either scenario, but I think the two alternatives have different advantages.

• Thanks for your answer, now I understand this a little better. To be honest, I kind of disappointed myself not noticing this difference, which - even in my language - is quite obvious :) – Michal Leszczyk Nov 18 '14 at 17:32
• It could also be taken as meaning "a bookie would offer you odds of a million to one" which, since bookies plan their odds so as to ensure they make a profit, would make the chances much less than a million to one. Of course as hyperbole, rather than literal odds, it amounts to the same thing. – Jon Hanna Jan 13 '15 at 17:24

a # to one is a common way of expressing odds or probability.

Another common way is to say there is a # in a # chance that something will happen.

For example:

The odds of winning that contest are a million to one.

You have a one in a million chance of winning that contest.

They are both correct and common. However, what complicates the matter is that "one in a million" (specifically one and a million, in that construction) means "special," "rare," or "somehow extraordinary." This meaning uses the probability expression simply to underline how special something is.

• When describing odds, N to 1 does not mean the same as 1 in N. They describe slightly different arithmetic. N to 1 is equivalent to 1 in N+1. Only when used casually, for hyperbole, are they effectively the same. – itsbruce Nov 18 '14 at 15:59

He is correct. Both one in a million and a million to one are two different ways of expressing a ratio. Mathematically, one in a million would appear as 1/1,000,000 and a million to one would appear as 1,000,000:1. Both are correctly written, correctly spoken and they mean the same thing.

• Strictly speaking, I think this is wrong. – itsbruce Nov 18 '14 at 15:50
• Yes? Why? definitely a legitimate possibility... – Gitty Nov 18 '14 at 16:12
• I have explained in full both in my other comments here and in my own answer. You chose to talk about the mathematics of the expression, rather than the casual meaning, and your mathematics are wrong. – itsbruce Nov 18 '14 at 16:19
• Yes I see what you are saying- it's 1/1,000,000 versus 1/1,000,001. When we are talking about numbers this large though, you must admit that the difference is rather trivial. – Gitty Nov 18 '14 at 16:52
• I did say "strictly" and I didn't mark you down ;) – itsbruce Nov 18 '14 at 18:37

Both usages are correct. Rusty Tuba is also correct that "one in a million" has a special meaning. However, these are definitely not equivalent statements mathematically.

"a million to one" is expressed in terms of odds, while "one in a million" is a direct ratio (1/1,000,000). So these two phrases imply different underlying probabilities.

Let's use smaller numbers to make sense of this.

"You have a one in two chance of winning that contest."

This means you will win 50% of the time.

"The odds against winning that contest are two to one."

This means you will only win 33% of the time.

Check it out with this odds calculator.

• Great point twosheds – Rusty Tuba Nov 18 '14 at 14:54
• You nearly have it right. Odds and probability are essentially the same thing (although Odds implies gambling). What is important is that "to 1" and "1 in" describe different mathematical results. – itsbruce Nov 18 '14 at 16:00
• @itsbruce: That's precisely what I was saying. The expression "n to 1" implies a different underlying probability than the expression "one in n." I've edited for clarity. – two sheds Nov 18 '14 at 16:21
• Yes, that is clearer. – itsbruce Nov 18 '14 at 16:23