Finally got round to knocking up a simulation, can confirm that the two methods (randomly choosing whether to bet H or T, or using the lagged counts) both give the same (0.5) win rate, as expected. Indeed, any strategy (e.g. always choosing H or T) will yield the same win rate for a truly random coin. I suspect there was a slight glitch in your friend’s simulation. Happy to share (R) code and plot(s) if you want.