🎰 Play the St. Petersburg Game
Flip until HEADS. Win $2n where n = number of flips.
H
T
Flip #0
$0
Bankroll: $1000
0
Games Played
$0
Avg Winnings
$0
Max Win
$0
Median Win
∞ Why Expected Value = Infinity
The Calculation
½ × $2 +
¼ × $4 +
⅛ × $8 +
1/16 × $16 + ...
= $1 +
$1 +
$1 +
$1 + ...
= ∞
| Flips | Probability | Prize | Contribution to EV |
|---|---|---|---|
| 1 (H) | 50% | $2 | $1 |
| 2 (TH) | 25% | $4 | $1 |
| 3 (TTH) | 12.5% | $8 | $1 |
| 10 | 0.098% | $1,024 | $1 |
| 20 | 0.0001% | $1,048,576 | $1 |
🤯 The Paradox
Each row contributes exactly $1 to expected value. There are infinitely many rows. So EV = ∞. But would YOU pay $1,000,000 to play once? Of course not! You'd almost certainly win only $2 or $4.
📊 Monte Carlo: What Actually Happens
Most wins: $2-$4 | Rare jackpots skew the mean
—
Mean (Average)
—
Median
—
Max Win
—
Most Common
"The median outcome is just $2. Half of all games end on the first flip! The infinite expected value comes from astronomically rare events."
🧠 Daniel Bernoulli's Solution (1738)
Utility Theory: The Birth of Modern Economics
Bernoulli argued we don't value money linearly. Gaining $1M when you have $100 is life-changing. Gaining $1M when you have $1B is a rounding error.
U(w) = ln(w) — "Log Utility"
With log utility, expected UTILITY is finite:
E[U] = Σ (½)ⁿ × ln(2ⁿ) = ln(4) ≈ 1.39
For someone with $1000, the "fair" entry fee using log utility is only about $4-5!
Historical Impact: This paradox led directly to:
- Expected Utility Theory
- Diminishing Marginal Utility
- Risk Aversion in Economics
- Modern Decision Theory