Risk of Ruin & Standard Deviation Explained

Standard Deviation: Measuring Volatility

Standard deviation (SD) quantifies how much individual results deviate from the expected value. A high SD means wide swings. A low SD means results cluster tightly around the mean.

For a single even-money bet (win $1 or lose $1) with 50% probability, the SD per bet is $1. After N bets, the SD of your total result grows with the square root of N:

This means doubling your number of bets does not double your risk — it multiplies it by about 1.41 (√2). The house edge, however, grows linearly. So the longer you play, the more the edge dominates the variance.

How to Use the Standard Deviation Calculator

1. 1.Enter win probability, win amount, and loss amount. These are the same inputs as the expected value calculator — they define the shape of the game.
2. 2.Set the number of rounds. Explore how the SD grows with √N but the expected loss grows linearly with N, showing why playing longer always favors the house.
3. 3.Read the result ranges. The calculator shows 1 SD (68% of outcomes), 2 SD (95%), and 3 SD (99.7%) ranges around your expected result. Even the optimistic end of these ranges eventually turns negative as you play more.

What Is Risk of Ruin?

Risk of ruin (RoR) is the probability that you will lose your entire bankroll before reaching a profit target. It depends on three things:

With a negative-EV game and infinite play time, the risk of ruin approaches 100%. The only question is how long you have before it happens.

How to Use the Risk of Ruin Calculator

1. 1.Enter your starting bankroll. The total amount of money you are willing to risk in a session, in dollar terms.
2. 2.Enter your bet size. The fixed amount wagered each round. A lower bet-to-bankroll ratio gives you more bets and more time — but the outcome remains the same.
3. 3.Enter the house edge. European roulette is 2.70%, blackjack with basic strategy around 0.5%, American roulette 5.26%.
4. 4.Set a profit target (optional). The goal you want to reach before stopping. The calculator shows the probability of hitting ruin before reaching this target.
5. 5.Compare scenarios. Try halving your bet size. Notice how RoR drops — not because you are safer mathematically, but because you have more bets before going broke. The expected loss is the same.

Key Takeaways

• •Standard deviation grows with √N. The house edge grows with N. Time always favors the casino.
• •A larger bankroll delays ruin but does not prevent it when EV is negative.
• •Betting smaller reduces variance and buys time — but the math is unchanged.
• •Risk of ruin is 100% in the long run for any negative-EV game with no stopping rule.