Every bet here has an edge. Some sizes still go broke.

Win probability just over half, every single time — a real, positive edge. Bet a small share of your bankroll and it compounds beautifully. Bet too large a share of the exact same bet, and the identical edge drives you to ruin almost surely, even though its average payout never stops climbing.

What you're seeing

Every round is the identical bet: win with probability p, lose with probability 1−p, at even odds. You choose what fraction f of your current bankroll to risk each round — a win multiplies it by (1+f), a loss by (1−f). Because each round multiplies the last, what matters long-run isn't the average dollar outcome but the average logarithm of it — and that quantity, the growth rate per bet g(f), is what the top chart draws. It starts at zero, rises to a peak at the amber line (the Kelly-optimal fraction f* = 2p−1), then comes back down and crosses zero again at the dashed red line. Past that second crossing, growth is negative — the identical positive-edge bet, sized too large, now shrinks your bankroll on average, every round.

The bottom chart makes that concrete: 300 gamblers, all playing the exact same edge at the exact same bet size, starting with one dollar each. Drag the bet fraction to somewhere modest — below or near the amber line — and both the mean (dashed cyan) and the median (solid amber, the typical gambler) climb together. Now drag it past the red ruin line and press Play again: the mean keeps rising, sometimes spectacularly, while the median peels off and sinks toward the bottom of the chart. Both lines are true. The mean is dragged upward by a shrinking sliver of gamblers who get improbably lucky and end up holding astronomical sums; the median is what almost everyone else actually experiences, and it is heading to zero.

Press Bet it all to see the extreme, cleanest version of the same fact. At f=1, one single loss — ever — empties the bankroll for good; surviving N rounds requires winning every one of them, so the survival probability is exactly p^N. At p=0.6, that's about 5×10⁻¹² percent still solvent after just 60 straight bets — yet the mean bankroll over those same 60 bets has grown roughly 56,300-fold, because the fewer and fewer survivors keep getting their stake doubled. Judging a bet by its expected dollar value alone never notices any of this: that naive multiplier climbs in a straight line all the way to f=1, with no peak and no warning.

The rule, exactly. Long-run growth per bet is g(f) = p·ln(1+f) + (1−p)·ln(1−f) maximized at the Kelly fraction f* = 2p−1 (no closed form exists for the second root, f_ruin; the page finds it the same way the verify script does — bisection). The naive "expected dollars" multiplier m(f) = 1+f(2p−1) is linear in f and never turns over. Verified in node (improve/verify/76-kelly-criterion.js, 60+ checks across seven values of p): an independent ternary-search argmax matches f*=2p−1 to within 1e-4 for every tested p; a 400,000-bet single trajectory's empirical per-bet growth matches g(f) in all four regimes (below Kelly, at Kelly, between Kelly and ruin, past ruin); an exact binomial-theorem identity confirms E[bankroll]=(1+f(2p−1))^N to six decimal places, including deep past ruin, where it keeps climbing (13.5× at bet 20 to 182× at bet 40, for one tested case) even as the population's typical outcome is collapsing. A 4,000-gambler ensemble at p=0.6, f=0.634 (past ruin) shows the median falling 4.71×10⁻³ → 4.92×10⁻¹⁰ → 5.85×10⁻³⁸ from bet 50 to 200 to 800, while the exact mean over the identical horizon climbs 3.90×10² → 2.31×10¹⁰ → 2.85×10⁴¹ — mean and median, same bets, opposite stories. The known-bad case, f=1: simulated survival after 12 all-in bets matched exactly p¹² for p∈{0.55, 0.60, 0.70}; after 60 bets at p=0.6, survival is 4.89×10⁻¹²percent while E[bankroll]=(2p)⁶⁰≈5.63×10⁴ — the mean never sees the ruin that almost every individual gambler is already in.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 76.