A biased coin game: heads doubles your capital, tails halves it. Starting capital: $100. The more rounds you play, the higher your expected winnings—but also the more certain your ruin.
Final Capital
$0.00
0% change
Expected Value
$0
Typical Outcomes (±1σ)
$0 — $0
Heads / Tails
0 / 0
Actual Heads %
0%
Note: Y-axis uses logarithmic scale to show exponential growth/decay
Expected growth per round: +0% — The expectation value suggests profits!
Typical growth per round: ≈ 0% — But in the long run, you typically lose many more rounds than you win.
The rare huge wins inflate the expected outcome, but the player typically goes broke.
The probability of winning an individual round (doubling the capital) is p < 1/2.
Expected capital after n rounds: E[Xn] = X0 · ((1 + 3p)/2)n → ∞
Typical capital after n rounds: Xn ≈ X0 · 2n(2p−1) → 0