Of ther above, Reverend Bayes' explanation is the clearest and most concise. Without any maths, either.

As for many of the skeptics mentioned in the original post being "logicians and engineers", if they were *good* logicians and engineers they'd do the experiment suggested by DMcG and see for themselves that they do win more often if they switch....then work on understanding the logic of the statistics.

I spend a chunk of my time doing statistical analysis of data to determine its precision and reliability forgeophysical and engineering purposes and it surprises me how often people who are otherwise great engineers have problems understanding statistics.