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I was told the following:

A presumption is something you think is true before you know any facts about the matter.

An assumption is something you think is true when you miss information, but you think you have it.

The difference can be subtle. When you have certain set ideas about some things, they are also presumptions.

Women can not drive cars is a presumption.

Based on the presumption, I can assume that you can not drive, because you are a woman.

Is this right, or is there more to it?

Based on the base word definitions:

as·sume

əˈso͞om

verb

suppose to be the case, without proof.

"you're afraid of what people are going to assume about me"

synonyms: presume, suppose, take it (as given), take for granted, take
as read, conjecture, surmise, conclude, deduce, infer, reckon, reason,
think, fancy, believe, understand, gather, figure

Versus:

pre·sume

priˈzo͞om

verb

suppose that something is the case on the basis of probability.

"I presumed that the man had been escorted from the building"

synonyms: assume, suppose, dare say, imagine, take it, expect,
believe, think, surmise, guess, judge, conjecture, speculate,
postulate, presuppose

So an assumption does not have any proof, whereas a presumption has some backing of proof in the form of probability or a "hunch".

More details can be found in this article.

Dictionary discussions of 'assumption' and 'presumption'

I found several treatments of these two words in different reference works. Although some of the discussions are clearer than others, they all generally agree that a presumption is more grounded in observed probability than an assumption is. The most concise discussion appears in Bryan Garner, Garner's Modern American Usage, second edition (2003):

assumption, presumption. The connotative distinction between these words is that presumptions are more strongly inferential and more probably authoritative than mere assumptions, which are usually more hypothetical. Presumptions may lead to decisions, while assumptions typically don't.

S.I. Hayakawa, Choose the Right Word (1968) doesn't cover presumption at all, unfortunately, but it does make the following point about assumption in contradistinction to principle, axiom, and theorem:

In logic, an assumption specifically designates the minor or second premise in a syllogism. Less specifically, assumption refers to any asseveration about reality which is unproved or debatable: [example:] the danger of basing scientific conclusions upon assumption.

The most extensive discussion of assumption and presumption appears in Merriam-Webster's Dictionary of Synonyms (1984), where it appears as part of a broader entry for terms relates to the verb presuppose:

presuppose, presume, assume, postulate, premise, posit are comparable when they man to take something for granted or as true or existent especially as a basis for action or reasoning. Their corresponding nouns presupposition, presumption, assumption, postulate, premise, posit when they denote something that is taken for granted or is accepted as true or existent are distinguishable in general by the same implications and connotations as the verbs. ... Presume and presumption may imply conjecture [example omitted] but ordinarily they carry the implication that whatever is taken for granted is entitled to belief until it is disproved. Therefore one presumes only something for which there is justification in experience, or which has been shown to be sound in practice or in theory or which is the logical inference from such facts as are known [examples omitted][.] Assume and assumption stress the arbitrary acceptance as true of something which has not yet been proved or demonstrated or about which there is ground for a difference of opinion [examples omitted][.]

And finally, Adrian Room, The Penguin Dictionary of Confusibles (1979) has this discussion of the verb forms assume and presume:

presume/assume (suppose take for granted)

'You'll come, I presume?': the implication is that I'm taking it for granted that you will, and I would be surprised if you didn't. 'You'll come, I assume?': the implication is that I am expecting you to because I have so decided or because it is your duty or obligation to, 'Assume' thus almost hints at an already completed action or precondition, while 'presume' relates to a simultaneous action or one in the future. To use 'assume' for 'presume' can seem presumptious, as well as being incorrect.

Famous quotations

One of the most famous instances of presume in the world of famous quotations is Henry Stanley's remark "Dr. Livingstone, I presume?" upon meeting—in a place where very few Europeans were about—the Scottish missionary/physician he had been searching for in the interior of Africa for some months. It is a classic instance where the speaker had good cause to expect that his implicit inference about Livingstone's identity was grounded in a strong logical probability.

An example of assume in its pure form occurs in G.H. Lewes, The Physiology of Common Life (1859):

We must never assume that which is incapable of proof.

[Entry listed in The Oxford Book of Quotations, third edition (1979), as is the Stanley quote from Henry Stanley, How I Found Livingtone (1872).]

The situation on the ground

How clearly do English speakers distinguish between the meanings of assumption and presumption? In the area of presupposition, I suspect, not very. Part of the problem is the array of complicating alternative meanings that the two words have. Here is the entry for assumption in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

assumption n (13c) 1 a : the taking up of a person into heaven b cap : August 15 observed in commemoration of the Assumption of the Virgin Mary 2 : a taking to or upon oneself {the assumption of a new position} 3 : the act of laying claim to or taking possession of something {the assumption of power} 4 : ARROGANCE, PRETENSION 5 a : an assuming that something is true b : a fact or statement (as a proposition, axiom, postulate, or notion) taken for granted 6 : the taking over of another's debts

And here is the corresponding entry for presumption:

presumption n (13c) 1 : presumptuous attitude or conduct : AUDACITY 2 a : an attitude or belief dictated by probability : ASSUMPTION b : the ground, reason, or evidence lending probability to a belief 3 : a legal inference as to the existence of truth of a fact not certainly known that is drawn from the known or proved existence of some other fact

The relevant definitions overlap and are crowded by other meanings, including one for each word that is far from complimentary ("arrogance, pretension" in the case of assumption, and "audacity" in the case of presumption). Under the circumstances, I doubt that most people who use the two words employ them in strict accordance with the distinctions laid down by Bryan Gardner, Merriam-Webster, or Adrian Room.

Still, if you want to align your usage with their recommendations, you should probably try to use presumption in settings where a fact-based inference is involved and assumption where a less firm conjecture provides the underpinning.

Assume seems to suggest making a judgement based on fact or information available, whereas presumption is to assume without proof, in the same way that prejudice is to pre-judge.

Oh, dear! The most up-voted response at present is so wrong as to be backwards. As other responses correctly state, one "presumes" when one has some basis for a position but not enough for certainty and one "assumes" when one has no knowledge and is just making a guess. "I presumed that you would again arrive late since you've never once been on time." "I just assumed that you would arrive late since everybody else does." Grammarly, Oxford Dictionaries and Merriam-Webster all back me up.

the root of the words is 'sume' which is to take or use. The pre and as are the prefix to that root. per means 'before and as is ad which it 'towards' Thus the two word are time dependent. One has history and one has future. presumption is guess based on knowledge whereas assumption is to pull it out of you hindquarters.