A Fair Coin, a Biased Coin, and the Odds of Doubling Before Ruin
Posted: Thu Oct 29, 2026 10:26 am
A Fair Coin, a Biased Coin, and the Odds of Doubling Before Ruin
Scenario: you start with 1,000 dollars and bet 100 dollars per flip until you reach 2,000 dollars or 0. What are your chances of doubling?
Fair coin. With a 50 percent chance on each flip, and the bankroll measured in units of 100 dollars, you start at 10 units and need 20. The chance of reaching 20 before 0 is the start divided by the target: 10 divided by 20, which is 50 percent.
Biased against you, 45 percent. Let r be q divided by p, which is 0.55 divided by 0.45, about 1.2222. The chance of reaching 20 before 0 from 10 is (1 minus r to the 10th) divided by (1 minus r to the 20th). r to the 10th is about 7.44, and r to the 20th is about 55.4. So the chance is (1 minus 7.44) divided by (1 minus 55.4), which is minus 6.44 divided by minus 54.4, about 11.8 percent.
A 5-point disadvantage per flip cuts the chance of doubling from 50 to about 12 percent. The ruin probability is the complement: about 88 percent.
Biased in your favor, 55 percent. r is 0.818. r to the 10th is 0.134, r to the 20th is 0.018. The chance is (1 minus 0.134) divided by (1 minus 0.018), which is 0.866 divided by 0.982, about 88 percent.
So a 5-point edge in either direction moves the odds from a coin flip to near certainty, because of repetition.
Lesson: with enough trades, small edges dominate. And costs, which turn a 50 percent game into a 45 percent game, are decisive.
Exercise: compute the doubling odds if you bet 50 dollars per flip instead, with a 45 percent coin.
Scenario: you start with 1,000 dollars and bet 100 dollars per flip until you reach 2,000 dollars or 0. What are your chances of doubling?
Fair coin. With a 50 percent chance on each flip, and the bankroll measured in units of 100 dollars, you start at 10 units and need 20. The chance of reaching 20 before 0 is the start divided by the target: 10 divided by 20, which is 50 percent.
Biased against you, 45 percent. Let r be q divided by p, which is 0.55 divided by 0.45, about 1.2222. The chance of reaching 20 before 0 from 10 is (1 minus r to the 10th) divided by (1 minus r to the 20th). r to the 10th is about 7.44, and r to the 20th is about 55.4. So the chance is (1 minus 7.44) divided by (1 minus 55.4), which is minus 6.44 divided by minus 54.4, about 11.8 percent.
A 5-point disadvantage per flip cuts the chance of doubling from 50 to about 12 percent. The ruin probability is the complement: about 88 percent.
Biased in your favor, 55 percent. r is 0.818. r to the 10th is 0.134, r to the 20th is 0.018. The chance is (1 minus 0.134) divided by (1 minus 0.018), which is 0.866 divided by 0.982, about 88 percent.
So a 5-point edge in either direction moves the odds from a coin flip to near certainty, because of repetition.
Lesson: with enough trades, small edges dominate. And costs, which turn a 50 percent game into a 45 percent game, are decisive.
Exercise: compute the doubling odds if you bet 50 dollars per flip instead, with a 45 percent coin.