In the world of Harry Potter, it appears that Divination is a mostly hereditary field; the only seers shown in the books are all related. Students studying the field in class, however, appear to have limited luck -- they try to see the future in leaves and cards, but most seem to go nowhere. Firenze describes the human version of Divination as  Self-flattering nonsense  The centaurs, however, use a form of Divination that is much closer to Astrology.  That form, too, appears to have limited success. Trelawney herself says  Many witches and wizards, talented though they are in the area of loud bangs and smells and sudden disappearings, are yet unable to penetrate the veiled mysteries of the future.  So the question remains: Can a student succeed at Divination via sheer hard work?
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In Order of the Phoenix, Dumbledore tells Harry  'I did,' said Dumbledore. 'On a cold, wet night sixteen years ago, in a room above the bar at the Hog's Head inn. I had gone there to see an applicant for the post of Divination teacher, though it was against my inclination to allow the subject of Divination to continue at all. The applicant, however, was the great-great-granddaughter of a very famous, very gifted Seer and I thought it common politeness to meet her.  Why did Dumbledore want to discontinue Divination? Is it because he didn't much like the branch of magic or didn't believe in it? It does not sound plausible that he would remove a subject from Hogwarts just because of his personal opinions. (Moreover, Divination examinations are taken during the O.W.L.s so I doubt whether Dumbledore would, with his individual decision only, be able to remove it completely from Hogwarts). From Order of the Phoenix we also know that there is whole room dedicated to storing prophecies in the Department of Mysteries, so prophecies and divination are a real thing in the Harry Potter universe, from the point of view of us readers as well as the characters themselves. Then why would Dumbledore want to remove Divination? Another thing to consider is that Professor McGonagall says, in the Prisoner of Azkaban, that true Seers are rare, so it might be that Dumbledore might have found it difficult to get someone competent enough for the job. This does not make much sense either, because if in the Harry Potter universe true Seers are rare, having someone teach the theory behind divination would at least help to keep the field alive. Also, he had trouble finding people for Defense Against the Dark Arts as well, but that does not mean he just gave up on the subject, rather, he made controversial hiring such as Professor Lupin and Moody to keep the classes running smoothly. What exactly was the reason for Dumbledore not wanting to continue Divination at Hogwarts (until he met Professor Trelawney)?
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A lot of the spells in Harry Potter, including just about every offensive one like Stupefy are what I would call "missile" spells (if there's a better term for it, please tell me). That is, they basically involve shooting a spell out of your wand like a missile, as opposed to an "area of effect" form (a protective spell like Muffliato) or no really perceivable form at all (like Accio).   Do such directed spells have a limit to their range? Or will they travel until they hit something?   I'm interested because if spells have virtually infinite range and high accuracy (which is another matter), then there should be "snipers" in battles that fire curses from long range, and it would be possible to "assassinate" somebody like Voldemort or Dumbledore given appropriate conditions. Of course, people may see the spell coming, but if they're distracted enough or if there's enough interference (a firework show, for example), then this could be a very effective means of killing people from afar.
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Something always bothered me with Fudge's accusations:     “Disciplinary hearing of the twelfth of August,” said Fudge in a   ringing voice, and Percy began taking notes at once, “into offences   committed under the Decree for the Reasonable Restriction of Underage   Sorcery and the International Statute of Secrecy by Harry James   Potter, resident at number four, Privet Drive, Little Whinging,   Surrey." - OotP Ch. 8 "The Hearing"   Harry cast the Patronus Charm, so he violated the Decree for Reasonable Restriction of Underage Sorcery alright - extinuating circumstances aside - but how did he breach the Statute of Secrecy?  The only witnesses were Dudley and Mrs. Figg, both of whom already knew that Harry is a wizard. Earlier, even Hermione gets it wrong (don't tell her!):     They haven’t even reported that you broke the International Statute of   Secrecy. We thought they would,... - OotP Ch. 4 "#12"   Yes, Harry does use magic in front of a Muggle, Dudley, but even Dudley is not so thick that the fact that Harry's a wizard would come as a revelation to him. Harry cannot breach the Statute of Secrecy to someone already in the know.  Shouldn't Dumbledore point out that glaring error in the indictment sooner rather than later? Or at least mention it, at all?  Is there maybe a source that mentions the Statute of Secrecy containing a clause, that even the mere possibility of magic being witnessed by an uninitiated muggle is sufficient for its breach?? Cause if so, wouldn't even adult wizards have a really tough time doing anything, ever.  Possibly related:
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One of the basic premises of the Potterverse is that Muggles are much, much more common than wizards. This is what in part necessitates the need for the Wizarding world to hide itself away because, despite their superior powers, they would be overwhelmed by the far more massive Muggle population should conflict break out.  However, we also know that wizards and witches have been around a long time. In Prisoner of Azkaban, Hermione's letter mentions:      I bet he’s learning loads, I’m really jealous – the ancient Egyptian wizards were fascinating.   Given that in one of Ron's letters, the Pyramids were mentioned as magical locations, this means that while we don't have a definitive date of when wizardry entered the world, we can definitively say that wizards and witches have been around since at least  Most likely they have been around much longer than that.  I could not find any evidence that suggests that magical people reproduce any faster or slower than a Muggle. Furthermore, being born to magical parents almost always means the child will also be magical (Squibs are considered very rare). While I don't have statistical evidence, it seems that merely having only one magical parent is enough to make it likely that the child will have magical abilities - based on Severus Snape, Seamus Finnigan, and of course, Tom Marvolo Riddle. Finally, there are of course Muggle-Borns, which while they are certainly very rare as a percentage of the Muggle population, they seem quite common enough at Hogwarts.   Finally, it's fairly easy to make the argument that Wizards are more powerful than Muggles in ancient times, given the lack of Muggle technology. In the modern era, there are  that Muggle weapons have exceeded Wizarding power, though there is not total consensus on this. It would be hard to argue, however, that Muggles had superior power (via technology) than Wizards prior to the 20th century.   In summary:   Magical people have been around for at least 4500 years, probably longer. Magical people reproduce at roughly the same rate as Muggles. For most of history, magical people have had superior power to Muggles.   The question:  Given these reasons, why is is it that the Wizarding population does not dominate the human population via ?  In-universe answer only, please. I'm not looking for a "J.K. Rowling is bad at maths" kind of answer.
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It is obvious that different wizards can cast spells with different power and/or effect (same for potions). But why?   Just some observations from my side - perhaps I'm wrong for some of them:  Spell casting seems to be like playing music or painting. More or less everyone can do it and the quality depends on some factors like:   knowledge - gained by studying experience - gained by practicing intelligence - innate and cannot be gained easily concentration at the moment (or ability to concentrate in general)   and last but not least   innate ability (talent) - you either have it or not   To me it looks like that the last one is determining everything. Some wizards like Dumbledore or Voldemort just "have it" and others will never be able to reach their power no matter how hard they try. Another example - Hermione is very intelligent and has read all the books but Harry beats her in defense against the dark arts nevertheless (and no - it's not Quidditch that helps him :) ).
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I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.  The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):  You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?  The character picks A, as the odds are all equally in his favor.  The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.  The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.  What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?  I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.  So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful.
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Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?  And even more, were they affected by the events happening in Europe?
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Students at Hogwarts are taught how to use spells, the correct incantation, pronunciation, tone of voice, wand action/movement, thought etc.  However Harry, who simply read the name of it in the Half-Blood Prince's Potions book, manages to produce the spell with no prior knowledge of how.  How does he know how to use Sectumsempra?
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In Harry Potter and the Half-Blood Prince, after Harry eavesdropped on Malfoy on the Hogwarts Express, Malfoy cast Petrificus Totalus on Harry while Harry was wearing his Invisibility Cloak.   How did the spell work, when Deathly Hallows tells us that no spell used against the Invisibility Cloak will work? (Harry's cloak was not an ordinary cloak and was a gift by Death, himself, to his ancestors, passed down to him)  To prove my point, sometime later in Deathly Hallows, one of the Death Eaters tried to Accio the cloak to find Harry, which didn't work.  So this contradicts J.K. Rowling theory.     Ah, but the Third Hallows is true Cloak of Invisibility, Miss   Granger! I mean to say, it is not a traveling cloak imbued with   a Disillusionment Charm, or carrying a Bedazzling Hex or else   woven from Demiguise hair, which will hide one initially but fade   with the years until it turns opaque. We are talking about a cloak   that really and truly renders the wearer completely invisible, and   endures eternally, giving constant and impenetrable concealment,   no matter what spells are cast at it. How many cloaks have you   ever seen like that, Miss Granger?
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In the Harry Potter world, when a magical person casts a spell, sometimes they target a specific object, like when Harry has a wand only and needs his broom, he casts "Accio Firebolt" and his broom comes to him.  But a lot of the time people point their wands just in the direction of someone to cast a spell.  One example would be in a duel.  When they do this, are they casting a spell on a particular person or at that person?  It seems, from what I remember of the movies, that some spells travel in a straight line while others specifically target a person or object.  Do any spells travel in one direction, like a laser beam?  What kind of spells are directional (instead of focusing on a particular person or object)?  And can directional spells be deflected or blocked by something like a mirror or another physical object?  (By "directional," I could have said linear, but didn't want to start sounding mathematical or scientific.  In other words, do they travel in one direction, like a laser beam would, or will they change direction to hit the intended target?)
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Is there some in-Universe (or somewhat canonical) explanation for why Harry wears glasses - either 100% of the time, or at least, when playing Quiddich?  Surely, given assorted spells, there should be one to repair either the issues with his near-sightedness, or, more likely, to act as lenses (being how magic is used to replace ANY technology, it ought to be able to replace lense-making).   P.S. I seem to recall a spell to make someone see better from afar, but not sure of the details.
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Are there either examples, or a mention of this having happened, of students permanently transferring from one magical school to another in Potterverse?  Canon only please (books/JKR/Pottermore).  Please note that I am only asking about transfers, e.g. you must already be studying in one school. Therefore an example of Malfoys considering sending Draco to Durmstrang is not in scope, since that happened before he attended Hogwarts. Also, guest-studying during Tri-Wizard doesn't count as it wasn't permanent.  Also, if the answer is a "no", please provide evidence (other than "well I read the books before and I don't remember a single case like that", unless you have eidetic memory) from entire canon.
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The spells that we know about in the Harry Potter world all seem to be either in Latin, derived from Latin, or resembling Latin.  This makes sense for the European World, but there were likely witches and wizards in Africa, Asia, and the Americas.  These cultures would have developed independently, and the magical people in them would likely have still found ways to cast their spells.  Do cultures with languages that did not derive from Latin use the same words and phrases that are related to Latin for their spells?  Or did they have other phrases based on the languages in their cultures?
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In universe, students are sorted into houses based on their qualities.  But when Harry Potter was sorted he was told, "He would do well in Slytherin."  The question is why?  There doesn't seem to be specific classes or instructors just for the specific houses.  While you could assume there would be mentoring or tutoring not in classes within each houses dorms, the fact that those students would have came up in the same house system and not have any different training other than that afforded all students.  Also the mentoring or tutoring isn't really acknowledged.  Why would any student develop differently in any house, other then being affected by like personalities and qualities?
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I always wondered if children who graduate from Hogwarts have an education that is actually worth while in the real world.  Do the children learn math and English?  I can see History of Magic as practical. I know of Arithmancy, but that's the study of the magical properties of numbers, not math.   Since Hogsmead is one of the very last all-magical towns, it's a fair assumption that a majority of the students at Hogwarts will be living in non-magical towns or cities. If practical classes (like math or English) aren't taught, then how do these students function in the non-magical cities?
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To my recollection, all physical Patronus charms that we see in the Harry Potter story are in the form of an animal. However, to quote  the nature of a Patronus charm is that it represents that which is hidden, unknown but necessary within the personality, which depending on the traits revealed could be embodied in many different ways.  For example, if a muggle-born wizard's Patronus were to portray strong leadership and perseverance could the physical manifestation appear as Optimus Prime?
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Leaving aside ethical and moral considerations, the whole policy of Salazar Slytherin's "we should not teach muggle-born magic-capable children" seems to be completely counter-productive to the whole "Let's not allow Muggles to know about existence of magical world" idea.  To wit, muggle-born wizards are obviously capable of performing magic (there were several examples in the book) even without wands/training, and many of them are powerful enough as far as their magical ability that it's likely to get noticed.   So, short of killing off every child that has latent magical abilities (or at least those with abilities beyond some threshold) or charming them so they are unable to do any magic, wouldn't refusal to teach them all how to PROPERLY use magic and what the rules are merely get a lot more chances for Muggles to see magic being performed which totally goes against the whole "Let's not allow Muggles realize that there are magic users around" that drives a lot of the Wizarding rules?  UPDATE Just to clarify - this question is about the idea of Salazar Slytherin himself and not the modern-day Death Eaters/Voldemort.