The real ingredients of Felix Felicis...Considering luck is really just a combination of probabilities, talents and intuition, examine how Harry acts after taking the stuff. Confident, determined, perceptive and charismatic. In light of the fact that the wizarding world and science are poles apart, they think it's magic. What is it really? A calibrated blend of Red Bull, alcohol and an opiate painkiller!
Slughorn chose the Drought of Living Death potion for the first day because...He knew the recipe in the book was substandard, and knew if anyone managed to pull the potion off properly instead of what you would get by following the instructions they would be deserving of the prize. He wanted a perfect potion - if the "best" in the class would have been merely passable, the prize would have not been handed out.
The recipe for the Drought of Living Death potion in Advanced Potion Making was dumbed down.So no students would make one that was actually dangerous. Not that there isn't plenty in the wizarding world at Hogwarts that could damage a person, but this particular author didn't want to contribute harm to anyone, or there are few remedies to this potion in its correctly-mixed form. After all, one would assume that Hermione - following the instructions as she does - would have passed this with flying colors.
The British Royal Family are all muggles.Harry is told "There are no wizarding princes".
In the film only: Dumbledore asked Harry about his relationship with Hermione because......at that time he was working on the deliminator for Ron (he knew he'd be dying and was hard at work on his will) and wanted to make sure Harry starting a relationship with Hermione would be off the table - because if that happened Ron may not have been able to use the deluminator to return.