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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?

kjetil b halvorsen
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Neil McGuigan
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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 08:26
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    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 03:28
  • @BrandonBertelsen: Because bigotry is funny? – naught101 Oct 22 '18 at 07:17
  • Personally I liked the first title "Your friend asks, "Hey how's a probability different than a plain old proportion?" Answer your friend in plain English". – Brandon Bertelsen Oct 22 '18 at 11:16
  • This question have much to share with the meaning of probability. The difference between frequentist and subjective approach is relevant here. This discussion can help https://stats.stackexchange.com/questions/447383/is-independence-subjective/447548#447548 – markowitz Mar 23 '21 at 13:43

8 Answers8

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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.

Jeromy Anglim
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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 08:38
  • 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
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    @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 04:51
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    @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
  • @Glassy The first three hits, from Ohio State & Harvard U, clearly distinguish "probability" from "proportion" (and favor the phrase "sample proportion" to emphasize the distinction). The third, in the Schaum's Outline series, refers to "an infinitely large binomial population with an unknown proportion p..." There are so many things wrong and vague about this we're best off recognizing that these outlines are not considered authorities. I suspect they may be trying to describe sampling in concrete ways: psychology research shows people reason better about proportions than probabilities. – whuber Aug 11 '11 at 15:22
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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.

Community
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whuber
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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
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    @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 00:03
  • @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 00:44
  • @whuber Ahh, it's population as an idea you're not happy with. Fair enough. (on the other hand it seems hard to talk about 'measurement subject to error' without that concept :-) – conjugateprior Jan 14 '11 at 08:35
  • @Conjugate Measurement refers to individuals, not a population. But that's taking us off on a tangent (for which I apologize): the question concerns distinguishing probabilities from proportions. I hope we have succeeded in illuminating some of the differences. – whuber Jan 14 '11 at 13:04
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    I meant that it's measurement *error* (or any other notion of statistical error) that requires the concept. But you're right, we've wandered a bit. Hope I'm not the only one who's been illuminated in this little exchange. – conjugateprior Jan 14 '11 at 14:37
  • I would say that a long run proportion is a hypothetical property while a probability is not, it is a statement about information. Say someone gives me a coin, tells me that it is *not* fair and flips it. What is the probability of the coin coming up heads? The hypothetical long run proportion is *not* 50% heads. Given the information I have I must still state that the probability of heads is 50% for this one flip as I haven't been told which way the coin is biased. – Rasmus Bååth Dec 10 '13 at 10:35
  • @Rasmus On the contrary, you ought to assert the probability of heads is *not* 50%. Lack of information should not be conflated with equiprobability. In a Bayesian setting you would elect to give a prior distribution for the probability, and you might select a prior whose *expectation* is 50% or even one that is symmetric around 50%, but you would have no basis for placing all its mass at 50%. – whuber Dec 10 '13 at 14:51
  • I think there may be an error in the "Ignorance is not probability" paper - shouldn't the 63% in "RA: I ran that in the simulation too and it came out to be around 63%." be **37%**, to be consistent with what comes later? – 410 gone Dec 10 '13 at 16:08
  • @Energy I appreciate your interest! The preceding sentence reads "What are the chances we could lose in all three?" I believe 63% is the answer to that rather than 37%. – whuber Dec 10 '13 at 16:13
  • But then later on you write: "DM acted under the assumption that she had better than even chances—63%, actually—of succeeding at least once," – 410 gone Dec 10 '13 at 16:16
  • @Energy I see. I think you are right about that--the subsequent text appears to be inconsistent. I am sorry the reviewers were not as sharp-eyed as you! I trust that this does not create any real confusion, though: it does not seem like a big enough error to warrant a correction. – whuber Dec 10 '13 at 16:19
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    No, no confusion - it just jarred as an inconsistency. It's a good paper - I enjoyed it. Re expert elicitation, you might be interested in [this paper from two colleagues of mine](http://dx.doi.org/10.1016/j.enpol.2013.06.110); though the data on the most amusing part, the calibration where energy experts were asked to put confidence intervals on their estimates of the length of the Moscow metro, didn't get reported. Let's just say Dunning-Kruger, and leave it there. – 410 gone Dec 10 '13 at 17:19
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    @Energy I wish it had been reported, because I'm sure the results were all over the place. It would reflect a situation--much like guessing oil prices in 2030--where the experts really have almost no valid applicable information. In that light their collective results about oil prices would look more confident and anchored in the present than they otherwise might seem. (I have modeled oil price fluctuations; the results provide ample reasons to be humble in making medium to long term forecasts.) – whuber Dec 10 '13 at 18:09
  • @whuber I'm making a distinction between the relative frequency/proportion of the coin which is not 50% and which I see as a hypothetical property of the whole process of flipping this specific coin and the probability of the coin coming up heads which is not a property of the coin but rather a property of my knowledge about the coin. I'm thinking this makes sence, no? – Rasmus Bååth Dec 11 '13 at 09:16
  • @Rasmus That was a mouthful ;-) but I think they are excellent distinctions. – whuber Dec 11 '13 at 14:24
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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.

Random
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The difference is not in the calculation, but in the purpose to which the metric is put: Probability is a concept of time; proportionality is a concept of space.

If we want to know the probability of a future event, we can use the probability at which the event took place in the past to derive our best estimate for the probability of the event in the future. If we want to know how much space is left in the theater then we use proportionality: the number of unoccupied seats/the number of seats.

This ratio is not the probability of securing a seat; the probability of securing a seat (a future event) is a function of the occupied and unoccupied seats, as well as the reserved seats, the no-show probability, and a myriad of other conditions.

Ferdi
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Doc
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    I see no reason whatsoever to tie probability to time, let alone future events. The fact that you have interesting and common examples here doesn't mean that you have identified the essential concept. – Nick Cox Nov 10 '17 at 08:59
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In the Frequentist Interpretation of Probability, the probability is long-run relative frequency. That means, the proportion in a large number of observations.

If you toss a coin 10 times and you get 4 tails, the proportion of tails is 0.4. However, 10 is not a very large number. This is only a proportion, not a probability.

Well, what is "a large number"? Bernoulli's estimation is 25,000 minimum. If you toss a coin around 50,000 times (for example) you are bound to get something close to 0.5 tails/heads. This means that the probability of getting heads/tails is 0.5. (and if you increase the number of trials, say, to 1 million, the proportion would become closer to 0.5)

The only problem here is: what is probability? Long-run relative frequency is how we MEASURE probability, but what is it? What does it really mean when I say "The probability of getting heads is 0.5"? This is one of the great problems with the Frequentist Interpretation. But this is my opinion only. I am sure many people differ.

Pluviophile
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Emre
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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

Nick Cox
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user35955
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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 '13 at 09:43
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    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 '13 at 09:46
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I don't know if there is a difference, but probabilities are not % they range from 0 to 1. I mean if you multiply a probability by 100 you get %. If your question is what's the difference between probability and % then this would be my answer, but this is not your question. The definition of probability assumes an infinite number of sampling experiments, so then we can never truly get probability because we can never truly conduct an infinite number of sampling experiments.

Brian
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