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Are Muggle-borns allowed to purposely cast spells using their wands before they go to Hogwarts?
If a Muggle-born had gotten their things from Diagon Alley but it is still before September first, then will the Ministry be able to detect them attempting to cast a Hover Charm for example?
Would they only be able to detect it if they got the spell right?
We only know of one such person, Hermione. She tried to practice some simpler spells before school (possibly on the train). We also do not know if she tried to cast anything (which could cause unwanted attention), or was merely trying to remember the wand movements and words of incantations.
Generally, a Muggle-born would be unaware that their strange abilities are magic (Harry releasing the big snake in the Zoo, strange things happening with Harry, Tom Riddle abusing his fellows, Lily Potter doing strange things with flowers). They only get an explanation when they are invited to Hogwarts.
The Ministry does not enforce any penalties on such use of Magic, but is probably aware of such children, as definitely is Hogwarts. The penalty for underage magic only applies once a child is accepted at Hogwarts and is explained that they would be punished for doing magic outside school, not before. Probably, such use of magic by an unaware child is not considered too dangerous.
Severus Snape explains to Lily Evans, a Muggle-born, that she wouldn’t get in trouble for the magic she’s done outside of school because she hadn’t gotten her wand yet and was young.
“… and the Ministry can punish you if you do magic outside school, you get letters.’
‘But I have done magic outside school!’
‘We’re all right. We haven’t got wands yet. They let you off when you’re a kid and you can’t help it. But once you’re eleven,’ he nodded importantly, ‘and they start training you, then you’ve got to go careful.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
That’d mean that once Hogwarts students do get their wands, they’d be watched more carefully, and therefore wouldn’t be allowed to purposely cast spells using their wands. However, it’s not clear how strictly this rule is enforced - for example, it may only really be enforced if they use magic around Muggles who aren’t allowed to know about it, but not if they only do it around their parents in their house. Hermione seems to have gotten away with casting a few spells before school.
“Are you sure that’s a real spell?’ said the girl. ‘Well, it’s not very good, is it? I’ve tried a few simple spells just for practice and it’s all worked for me. Nobody in my family’s magic at all, it was ever such a surprise when I got my letter, but I was ever so pleased, of course, I mean, it’s the very best school of witchcraft there is, I’ve heard – I’ve learnt all our set books off by heart, of course, I just hope it will be enough – I’m Hermione Granger, by the way, who are you?”
- Harry Potter and the Philosopher's Stone, Chapter 6 (The Journey from Platform Nine and Three-Quarters)
It doesn’t seem like Hermione got into any particular trouble for it either, as she seems quite happy to mention her attempts at spellcasting to students she just met - which she probably wouldn’t be if she’d gotten in trouble for it.
This doesn't seem to be addressed explicitly in the books. However, we do see two occasions where wizards raised amongst Muggles are informed of their wizardness, and in neither case are they told that they are not allowed do magic before going to Hogwarts. Hagrid did not tell this to Harry, nor did Dumbledore tell it to Tom Riddle.
Now it is possible that they didn't mention it yet it is still a rule, or it is possible that they mentioned it "off-screen" (although in both cases it does seem as if we are seeing the entirety of the interactions). Moreover, Dumbledore at one point explicitly warned Tom about certain rules and even then he did not mention a rule against using magic:
"At Hogwarts," Dumbledore went on, "we teach you not only to use magic, but to control it. You have — inadvertently, I am sure — been using your powers in a way that is neither taught nor tolerated at our school. You are not the first, nor will you be the last, to allow your magic to run away with you. But you should know that Hogwarts can expel students, and the Ministry of Magic — yes, there is a Ministry — will punish lawbreakers still more severely. All new wizards must accept that, in entering our world, they abide by our laws."
Additionally, in Harry's first Potions class we find the following:
"Thought you wouldn't open a book before coming, eh, Potter?"
Harry forced himself to keep looking straight into those cold eyes. He had looked through his books at the Dursleys', but did Snape expect him to remember everything in One Thousand Magical Herbs and Fungi?
This indicates that it was expected for students to do some form of magical learning on their own, even prior to ever coming to Hogwarts. This is not dispositive, though, as it is possible they were only expected to do things that would not involve the actual performance of any magic.
Furthermore, the Hogwarts letter that Harry received also made no mention of such a rule. And finally, Hermione seems to have used magic before school, and she is hardly the rulebreaking type:
"Are you sure that's a real spell?" said the girl. "Well, it's not very good, is it? I've tried a few simple spells just for practice and it's all worked for me.
In sum, there are a few indications that it might be allowed, but I don't think there is any explicit proof.
