If I understand the Wilson confidence interval correctly (and I'm not confident that I do), it doesn't seem to take any data into account other than the success/failure counts of the particular thing being tested. This seems kind of weird to me. For example:
If you flip a coin three times, and it comes up heads all three times, the Wilson confidence interval seems to say that there's a 95% chance that the real chance of this coin coming up heads on any individual flip is at least 43.8% (or something like that), which feels pretty reasonable to me.
But if a baseball player makes it up to the big leagues, and in his first game, he gets three hits in three at bats, it seems to say that there's a 95% chance that the real chance of this guy getting a hit in any given at bat is at least 43.8%. That is clearly not a reasonable conclusion.
Am I understanding this correctly? If not, how not? And if so, is there something similar to the Wilson confidence interval that takes into account typical success rates? For example, yes, we know that this guy got three hits in three at bats, but we also know that overall, major league batters get hits in something like a quarter of their at bats.