[jtr]

Just saw your response -- no offence, but maybe the reason for your discomfort is that you're not totally clear on the "95%" part of the confidence interval.

The 95% interval and its complement, the 0.05 alpha-level in statistical testing, is based on a throwaway line by Ronald Fisher, the guy who put the basics of modern statistics together, that this might be a reasonable starting point for statistical hypothesis testing. In many areas of science but particularly in psychology, the 5% / 95% standard has come to be seen as something set in stone rather than the arbitrary cutoff point it really is.

If you want to calculate some confidence intervals vis a vis your own play that will restore your faith in the mathematics, try switching "95%" for "99.9%" (closer to most people's idea of "just about a dead certainty" anyway). Then you will find that the intervals are wide enough to keep you happy.