Pr(same side)=0.508, 95% credible interval (CI) [0.506,0.509]

Toss it 100 times, overstating the effect you'd expect it to land on the same side it started 51 times, opposite side 49.

This seems to have been lost in much of the discussion. Employing this in professional NBA basketball you /might/ get one extra toss win per season out of your 100 games compared to any other way of selecting without taking into account the starting side.

Good luck using this!