At sixteen, I knew I had a shot at the 1976 Olympics. (Para.6)
Meaning: At sixteen, I knew I had a chance to take part in the 1976 Olympics.
have a shot at: attempt to do sth.
I doubt whether I can solve the problem, but I'll have a shot at it. 我怀疑我能否解决这个问题，但我会试试看。
Why don't you have a shot at talking to her yourself? 为什么你不尝试着自己与她谈？