The astrologer's belief in fate is evident in the way he interacts with his clients. He listens attentively to their stories and uses his astrological knowledge to provide them with advice and guidance. He believes that his clients' lives are already written and that his predictions are merely a way of revealing what will happen.
However, the story also suggests that fate is not entirely fixed. The astrologer's predictions are not always accurate, and he sometimes doubts his own ability to see the future. This suggests that fate may be malleable and that the actions of individuals can have a significant impact on their lives.
Ultimately, the role of fate in "An Astrologer's Day" is complex and ambiguous. The story suggests that fate plays a role in our lives, but it also leaves room for individual agency and choice.