We need the geometric rather than the arithmetic or median expected value. HH: 2.25 HT: 0.9 TH: 0.9 TT: 0.36. Expectation: (2.25 * 0.9 * 0.9 * 0.36)^(1/4) = 0

We need the geometric rather than the arithmetic or median expected value. HH: 2.25 HT: 0.9 TH: 0.9 TT: 0.36. Expectation: (2.25 * 0.9 * 0.9 * 0.36)^(1/4) = 0.9. So you expect to lose 10%. The losses don't get offset by the wins. Intuition: Possibility of losing 100% is not offset by a possibility of winning 100%, so the arithmetic mean is wrong.