Is this normal?

This is the first time I have been the one doing the hard work with potty training, when I worked at the day care we would work on it but they stayed in diapers until they were almost done. Ok so she always starts to potty in her underwear and then runs to the bathroom and finishes up. Is that normal? Just now she pee peed in her underoos and then ran on her own to the potty chair. I heard her and went in the bathroom and she had finished going. So I made a big deal about her going and only told her quietly that she needs to try and and get to the potty chair before she starts to go. I am not trying super hard though to train her, this is all her doing. I decided this weekend that I am going to buy cheap diapers though since she takes them off as soon as she wets them. She does recognize though when she has started to pee and then tells me. Maybe it is just a matter of controlling those muscles better and will get better with time. That what I was wondering if this is kind of normal when they are first starting out. As a child I hard a very hard time because my muscles weren't that strong and I had small kidneys so it caused me to have infections a lot.