I am a dog lover. A dog person. My parents bred sharpei and scottish terriers my entire life. I entered college and got a dog of my own almost immediately. I haven't been without a dog until the past couple of years with two small children and too little time to give a dog the attention it deserves. I am doing research on human animal bonds and how they change human perception!
With all of that said, dogs are animals. Dogs can bite. Sometimes they mean it, other times, it's an accident. I have a scar on my nose from when I was three and one of our older, aging scotties bit my nose in an effort to get at a ball. He didn't mean it, but he did it.
I have a scar on my leg where one of our sharpei's attacked me as I jumped over our backyard fence when I was locked out (as a teenager). He MEANT it.
I have been bitten countless other times by our own pets. The fact of the matter is that they do bite because they do not understand that it is "wrong" to bite for Pete's sake. They are either following a stimulus response pattern or they have an accident while playing and get too rough. It is the nature of the beast.
I can't tell you how many people have told me "oh, he's been my pet for years. He would never bite" I take it with a grain of salt and watch my children ultra closely around any dog. Occasionally a playdate will have a dog lose and I will have to, in some way, whether through body language or verbally ask for the dog to be put up. children are too wild sometimes. IT is not the dog's fault. Nor is it the fault of the child, depending on their age and understanding. Dogs must be treated with respect. I spent a palydate several weeks ago holding a small schnauzer mix by it's collar because the owner felt "comfortable" with the dog out. Fine, but it's not getting out of my reach. and it didn't The next time over, she had the dog put away.
As for your daughter, she is behaving irresponsibly,. However, as you know, she is showing her age. She still isn't quite capable of understanding the gravity of the situation. You are her mother and that child's grandmother. You have the right to at least guide her on what she must do if she loves her child. The dog doesn't need to be put down or anything horrible like that. But she does NEED TO KNOW that the dog is up-to-date on shots. The likelihood is that it is vaccinated. However, she still needs to know that.
She also needs to take the baby to the pedi and have the bite checked for infection. While a dog's bite is actually cleaner than a human bite, it can still leave room for infection, if even because it is an open wound.