Author's Comment: In my opinion, there are two most important things (for an adult to say to a young person). One is: "I was wrong." Any variation of this will do, such as "I didn't know," "I should have known," "I wish I had known," and of course, "My mistake." The significant message is that adults make mistakes, (because nobody is perfect and nobody knows or understands everything), and that the adult response to an error is to acknowledge it, directly.
There is no loss of face, or authority, in owning up to making an mistake. In fact, it is people who can never admit to doing anything wrong whom it is difficult to trust, in whose wisdom I have no confidence.
Two follows directly and logically from One and is: "I'm sorry." A truly grown-up person can understand that she/he is flawed and makes mistakes. There is no need to kid yourself about being perfect, is there? And aren't you sorry that you did whatever harm you did, when you were making your mistake? An apology is definitely in order.