Its normal, dont worry. What we did when my daughter hits (she hasnt done the biting) is that we stop her say NO very firmly, take care of the person hurt, and get my daughter and take her over to the person, get her hand and have her stroke whatever part she hurt and say, See gently, it hurts when you hit/bite someone and makes them sad. Do you want to say sorry and have cuddle?
It worked wonders for us as I didnt see the point in hitting her, and at the same time we showed her an acceptable way of treating people, consequences, and apologies. It might take a while, but it will work eventually. If it can work on my very very strong willed stubborn little girl, then it can work for anyone!