"Believe" in luck per se? No. Recognize that circumstances almost entirely beyond your control could be described as "lucky" or fortunate? Absolutely. People don't do this enough, thinking themselves masters of their own fate. For people to do otherwise is often scary to them, because it implies a loss of control that people don't like to think about.
Consider the circumstances of your birth, something none of us has any control over. When were you born? If it's this or last century your odds of having a better life are much better than being born several hundred or thousands of years ago. Where were you born? Your life is much more comfortable and stable of you were born in the West than say, Syria or Pakistan. What other circumstances were you born into? A middle class or wealthy family? Then you'll have chances others may never receive. Same with skin color. Have a great work ethic? Blessed with intelligence? Attractive or ugly? Between upbringing, psychology and genetics, how much credit can you really take for those? One could even have almost everything breaking their way except for one lousy determining factor which could change everything. Many people were lucky in that they were born into a middle class family that had good employment, good medical insurance and into the wealthiest country in the world. Then their kid got an awful illness or had a bad accident when they were young, decimating the family finances and causing a lot of pain and suffering that is born by the entire extended family.
Luck, good or bad. It happens to everyone. There's no need to believe in it, it just is a description of your personal circumstances. One thing it does to those that realize how influential it is - it increases compassion. It makes the idea of pulling yourself up by your own bootstraps ring somewhat hollow.