Why positive expected value doesn't save you
The bet has positive expected value. Each coin flip multiplies your wealth by 1.5 (heads) or 0.6 (tails), for an expected multiplier of 1.05 — a 5% gain per round. An economist averaging across many players at a single moment would call this a good bet.
But run it many times in sequence, and almost everyone goes broke. The ensemble average — the thick gold line — grows steadily because a handful of extraordinarily lucky players inflate it. Meanwhile the median player, and nearly every individual trajectory, collapses toward zero. The few survivors don't compensate for the many ruined; they mask them.
This is the difference between ergodic and non-ergodic processes. In an ergodic system, the time average equals the ensemble average — what happens to the group predicts what happens to you. Life is non-ergodic. You live one trajectory, not the ensemble. You cannot eat the expected value.
The key insight: for multiplicative dynamics, what matters is the geometric mean (time average), not the arithmetic mean (ensemble average). The geometric mean of this bet is √(1.5 × 0.6) ≈ 0.949 — a 5.1% loss per round. The arithmetic mean says grow. The geometric mean says die. Both are correct. Only one is yours.