# What's the difference between a probability and a proportion?

Say I have eaten hamburgers every Tuesday for years. You could say that I eat hamburgers 14% of the time, or that the probability of me eating a hamburger in a given week is 14%.

What are the main differences between probabilities and proportions?

Is a probability an expected proportion?

Are probabilities uncertain and proportions are guaranteed?

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I'm just wondering whether the edited version of this question should retain the aspect of the original question regarding how the distinction between probabilities and proportions could be described in lay terms. –  Jeromy Anglim Aug 11 '10 at 8:26
If you eat Hamburgers every Tuesday, the probability of you eating a hamburger in any given week is 1. –  Brandon Bertelsen Aug 12 '10 at 3:28
good discussion on here. thanks guys! –  Neil McGuigan Aug 13 '10 at 17:44
Personally I liked the first title: "Your blonde girlfriend asks, "Hey, how's a probability different than a plain old proportion?" Answer her in plain English." –  Brandon Bertelsen Aug 17 '10 at 1:58

I have hesitated to wade into this discussion, but because it seems to have gotten sidetracked over a trivial issue concerning how to express numbers, maybe it's worthwhile refocusing it. A point of departure for your consideration is this:

A probability is a hypothetical property. Proportions summarize observations.

A frequentist might rely on laws of large numbers to justify statements like "the long-run proportion of an event [is] its probability." This supplies meaning to statements like "a probability is an expected proportion," which otherwise might appear merely tautological. Other interpretations of probability also lead to connections between probabilities and proportions but they are less direct than this one.

In our models we usually take probabilities to be definite but unknown. Due to the sharp contrasts among the meanings of "probable," "definite," and "unknown" I am reluctant to apply the term "uncertain" to describe that situation. However, before we conduct a sequence of observations, the [eventual] proportion, like any future event, is indeed "uncertain". After we make those observations, the proportion is both definite and known. (Perhaps this is what is meant by "guaranteed" in the OP.) Much of our knowledge about the [hypothetical] probability is mediated through these uncertain observations and informed by the idea that they might have turned out otherwise. In this sense--that uncertainty about the observations is transmitted back to uncertain knowledge of the underlying probability--it seems justifiable to refer to the probability as "uncertain."

In any event it is apparent that probabilities and proportions function differently in statistics, despite their similarities and intimate relationships. It would be a mistake to take them to be the same thing.

### Reference

Huber, WA Ignorance is Not Probability. Risk Analysis Volume 30, Issue 3, pages 371–376, March 2010.

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Err, maybe I'm missing something but in some important cases, e.g. all of survey research, probabilities are not at all hypothetical, they're just population proportions. In the question 'how many Ukrainians think X' the population is pretty clear - all Ukrainians - and the proportion that thinks X from a simple random sample estimates the proportion of the population that thinks X, which is exactly the probability of interest. For frequentists, this is the easy case (and I, as a non-frequentist would concur with their analysis). –  conjugateprior Jan 13 '11 at 22:46
@Conjugate In some cases a probability may equal a proportion but it is not a proportion. What relates a proportion to a probability is the specific procedure of sampling uniformly at random with replacement from a well-defined population (which are rare, by the way: 20 Ukrainians have been born since you wrote your comment!). This clearly is a special case of other sampling methods, including without replacement, with stratification, etc. In those other cases the proportions no longer even equal the probabilities. Doesn't this suffice to show the two concepts are distinct? –  whuber Jan 13 '11 at 23:05
On the 'special case of other sampling methods', I agree. The whole trick of survey research is to make one's stratified, clustered, etc. sample proportions match (in expectation at least) the population proportion one is interested in. But I'm not sure that's a criticism, rather a statement of the problem. –  conjugateprior Jan 13 '11 at 23:56
Another tack: whenever a frequentist states a coverage probability she immediately asserts the existence of a population. In some situations the population is indeed infinite, e.g. infinite hypothetical replications of an experiment that takes place once. But in others, e.g. the survey context, the population is not infinite at all. In my example it's the population of the Ukraine (at t). That's definitely finite and describable using proportions. I'm not disagreeing with your analysis - just pointing out that there is a simple situation where proportions and probabilities coincide. –  conjugateprior Jan 14 '11 at 0:03
@Conjugate I am making a conceptual distinction here, so "coincidence" (mathematical equality), although noteworthy, is beside the point. In many cases, a "population" is a convenient fiction: one can find many references to processes as "populations," for instance. In ideal cases we can indeed construct a rigorous data frame. But that solves only one of the difficulties. Whatever we're interested in has to be measured and measurements are subject to error. (Even gender!) The ontological status of a "population" therefore is questionable: like probability, it too is a modeled construct. –  whuber Jan 14 '11 at 0:44

If you flip a fair coin 10 times and it comes up heads 3 times, the proportion of heads is .30 but the probability of a head on any one flip is .50.

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+1 for proportion is empirical, and it is often a good estimate of a probability which is theoretical ! –  robin girard Aug 11 '10 at 8:38
You change the viewpoint here. You could just as easily say, "the proportion of heads on any one flips is .50". I contend that probabilities and proportions are essentially the same. –  Neil McGuigan Aug 16 '10 at 18:13
@Neil I can see how the proportion of heads in one flip can be 1.0 or 0.0, but I cannot see how it ever can be 0.50 (except in a Schrodinger Cat experiment, perhaps, but that's a different issue...). –  whuber Nov 24 '10 at 4:51
@Neil : No you can't. It doesn't even make sense in regular english, let alone in statistics. –  Joris Meys Nov 24 '10 at 10:26
I agree with Robin, Anyhow, even if it is not usual to say that in a given set of observations the probability of success is 0.3, it is common to use the word proportion as a synonym of probability: search google for: binomial and "proportion p of success" –  glassy Jan 13 '11 at 13:56
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A proportion implies it is a guaranteed event, whereas a probability is not.

If you eat hamburgers 14% of the time, in a given (4-week) month (or over whatever interval you based your proportion on), you must have eaten 4 hamburgers; whereas with probability there is a possibility of having eaten no hamburgers at all or perhaps eaten a hamburger everyday.

Probability is a measure of uncertainty, whereas proportion is a measure of certainty.

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Proportion and probability, both are calculated from the total but the value of proportion is certain while that of probability is no0t certain..

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From my point of view the main difference between proportion and probability is the three axioms of probability which proportions don't have. i.e. (i) Probability always lies between 0 and 1. (ii) Probability sure event is one. (iii) P(A or B) = P(A) +P(B), A and B are mutually exclusive events

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Proportions mimic all three properties with corresponding properties of their own. Proportions (in the sense intended in the question) lie between 0 and 1, the proportion of times a sure event occurs is 1, and the proportion of time A or B occurs is the sum of the proportions if the events are mutually exclusive. –  Glen_b Dec 10 at 9:43
I am with @Glen_b. Not only are your claims not true, you don't even offer an argument on why they are true. Sorry, but your answer can't help anybody. –  Nick Cox Dec 10 at 9:46