Bill was sure that he knew the mouse ate the cheese. This is because, as he was sitting at the party he saw with his own eyes a “small mouse creep shyly over to the cheese and begin devouring it” immediately providing quite concrete evidence. Then, when he checked again -in case he was imagining it- the mouse, through his eyes, was still present eating the cheese. Also, to add to the reality of Bill seeing it with his own eyes, the passage confirms that he was not dreaming, drunk nor hallucinating.
I think Bills reasons for being sure that he saw the mouse eat the cheese are good reasons because, he literally saw the mouse eating the cheese with his own eyes which is a very reliable source of evidence. Prior to this, there is no implication that Bill would “want” to imagine a mouse, that he had bad eye sight and had mistaken something else for a mouse and to add to this, he also confirms that he is neither drunk, dreaming nor hallucinating.
Virginia and Adrian claimed to know that the mouse ate the cheese because Bill had told them the mouse had using the words, “Believe me. It’s a fact”. They both appeared to accept Bills logical explanation and they appeared to have no external reasons or desires to doubt Bills statement. Virginia could tell that Bill “obviously wasn’t joking” and because she trusted and knew him well enough she accepted his evidence. Adrian also trusted Bills evidence as they were “old friends” and had no need or reason to doubt Bill.
I think that both Adrian’s and Virginia’s reasons for claiming to know that the mouse ate the cheese are adequate. Their reasons to not provide concrete evidence as their reasons are second hand beliefs from someone else. However, because they both know Bill well and have reasons to trust his evidence and not doubt it, I believe their reasons are definitely at least adequate.