Harry Potter Questions :: Why are muggle born underage witches and wizards allowed to do magic?

Why isnt there any magic supervision for the muggle-born children?
I just thought of all the fuss everyone makes when Dobby performs the Hover Charm in Harry's house and when Harry performs the patronus. I also remember Dumbledore telling Harry how magic is NOT detected by underage wizards if an adult wizard is around.

However, to read an excerpt from Harry Potter and the Philosopher's Stone -
"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 best school of witchcraft there is, I've heard - I'e learnt all our set of books off by heart, of course, I just hope it will be enough - I', Hermione Granger, by the way, who are you?"
[I really enjoyed typing the above. It was like Hermione was speaking all this rather slowly as I typed it away.]

Well, in the second monologue that our Hermy every delivers, she clearly mentions two things -
Firstly, she tried a few spells and THEY ALL WORKED.
Secondly, no one in her family is magic, that is, there are no witches or wizards in her family.

Hence, now, I would like to know - why is Hermione allowed to do magic?

Considering the statute of secrecy (or whatever the Ministry has made of it), I really believe that magic done by a muggle born, unsupervised, is way more dangerous.
Take for instance, the Weasley twins - I am pretty sure that they have done underage magic many times. Yet, having seen their elder brothers, parents and other relatives do it so often, I am sure even if they did it all by themselves, they were less a threat to the world as a muggle born who was doing something no one around them knew of. Further, since Fred and George come from total wizarding family, surrounded by other wizards, even if they perform magic outside their houses, no one is going to look astounded. On the other hand, if Hermione performs magic inside her house, her dentist parents will look.

May have been the case that I might have missed something, but I would just want the Ministry of Magic to look into this matter seriously (not that we are not accustomed to carelessness on their part).

Snippets from Professor McGonagall's desk #2

Quirrell took out his wand and was performing complex powerful spells to put the troll into an enchanted sleep before tying him up. Unable to believe the last few moments, she irritably told Quirrell to tie the troll first and then continue. 'it was elementary, Said Snape irritably. 'I believed the defence against dark arts professor should know that' he continued with pauses 'the repurcussions of our choices, Quirinus.' Just as he finished, Albus Dumbledore entered the girls bathroom, closely accompanied by the ghost of Gryffindor tower - Nearly Headless Nick. On seeing the other teachers, he turned to Nearly Headless Nick, the shine back in his deep blue eyes, and said 'Thank You sir Nicholas. Your help has been instrumental. I would not like to keep you from all the celebrations anymore.' Nick smiled as he turned and glided away.
Dumbledore went to the the troll who had thick ropes tied all around him. Dumbledore waved his wand in random motion on the troll as if trying to shake off something sticking from it. Nothing happened. After a moment or two, he turned to Quirrell and asked with utmost astonishment 'I don't detect any magic. How did you stun the troll?' Snape let out a grunt and Quirrell stammered before beginning his sentence. Dumbledore turned to McGonagall and she said 'It was Potter, Weasley and Granger. They knocked him unconscious.' 'That is something, considering we are talking about three first years taking on a full grown mountain troll' smiled Dumbledore. 'Three arrogant and rule breaking first years, headmaster' reminded Snape. 'Yet impressive Severus. I am sure you would agree.' Snape grunted again. 'I am not sure you understand the situation here, headmaster' pleaded McGonagall. It appeared she was talking on something that alone she understood in the room. When no one said anything, she continued 'A full grown mountain troll! They pursued a full grown mountain troll. And he flies just like James, maybe better. Potter has unbelievable resemblance to James. I don't want another generation of Marauders in the castle, especially not when how they ended.' McGonagall's voice was trembling. It hinted anger and worry. Snape seized the chance and added 'Headmaster, he has inherited arrogance and rule breaking as well.' Dumbledore smiled kindly at them both 'What worries you Minerva is a boy taking after his father?' 'What worries me headmaster is our efforts for the past decade to keep the boy and our world safe. The Marauders didn't soar for long enough and these three have every making of a marauder' this time, it was just worry that reflected in her voice. 'The Marauders remain, Minerva. As you and I discuss of them, they are very much present. Now Severus, help Professor Quirrell show the door and if possible the borders of England to the troll.' Snape grunted again before he left with Quirrell, both were making the now tied troll floating behind them. Dumbledore turned to Professor McGonagall and then both of them started walking away from the girls bathroom. Enjoying the festive mood, Dumbledore was humming to himself. In the front of the stone gargoyle, Dumbledore stopped to say farewell to Professor McGonagall but she turned away, a little angry and irritated, muttering to herself 'I already had the Weasley twins and now another likely generation of Marauders. To top it all, they keep looking for chances to hurt themselves. I wonder why people think bad of the Slytherin house, try coming to Gryffindor. And I also deserve a raise with so much to manage.' Dumbledore heard no more as Professor McGonagall had turned the corner at the end of the corridor. He resumed his humming after saying 'Pumpkin Pasties' to the gargoyle.

Snippets from McGonagall's Desk #1

McGonagall was looking out of the window from behind her desk, pondering over the best way to introduce advance transfugations to her third years students without giving Fred or George Weasley any new idea to cause mayhem. She had almost decided to stick with strict-reading-from-the-book approach (the Weasley twins never bothered to read a word from their books; only Merline knew how they managed to pass every year!) when she saw something come zooming near her window. McGonagall let a small scream thinking it was a big owl or an Eagle gone astray. However the creature stopped as swiftly and effortlessly as possible. Like a reflex action, McGonagall said "James Potter again!" Her voice was filled with anger and fatigue which was replaced by sadness when she realised what she had just said. "It must be Harry." She said in a doleful tone. It was indeed Harry on the broom. He held something in his hand which McGonagall could only make out since the object was shining in the bright Sun. She saw Harry fly the broom back to the ground to his fellow classmates. There was no doubt that the boy flew as ably as his father if not better. Madam Hooch was not present anywhere to be seen and neither were any of the other students mounted on their brooms. McGonagall now had another dilemma to solve - having Harry on the Gryffindor Quidditch team would mean that Gryffindor would have the best seeker it has had since James left Hogwarts. It would also mean having the best seeker in all the four house teams. However, she couldn't let Harry become just like James, not when she knew how James ended up. As McGonagall looked down, she could see Harry standing with Ronald Weasley arguing with Draco Malfoy. That instant she knew that the genes were passed very strong, better aim for the Quidditch Cup. As she walked out of her room, towards the grounds, she let out "Bless you Hogwarts!"