Did you see _ white rabbit?oh i just saw _rabbit in a white coat passing by填冠词a或the

Did you see (the) white rabbit?
Oh,I just saw (a) rabbit in a white coat passing by.
你看见了那只白兔没有?
我刚刚见到一只穿白色外套的兔子经过.
第一句问的是特指的兔子；第二句则表示是一只兔子,是泛指.第一个问不该是泛指？第一空应该是体现出专门来寻找特定目标的，而不是随便问问哪一只。