[AaronBrown]

A man named William Gossett did this about a century ago. He wrote under the name of "Student" (he worked for Guiness Beer and they didn't want the notoriety of having a statistician in their breweries), so the result is called the "Student's t" distribution.

However, the correction is not very large if you have played a large number of hands and aren't asking questions about much larger samples of future hands. While uncertainty about your win rate is important, there are much more important factors like nonconstant win rate and rare events that make the computation less accurate than theoretical bounds suggest.