Outcome Distribution Calculator

Ever wondered what the full range of possible outcomes looks like for your gambling session? This calculator visualizes the probability distribution of results, showing you not just the expected outcome, but the entire spectrum of possibilities—from best-case to worst-case scenarios—and the likelihood of each.

How Outcome Distribution Works

When you make repeated bets, your total outcome follows a predictable statistical pattern. According to the Central Limit Theorem from probability theory, the sum of many independent random outcomes approaches a normal (bell curve) distribution—regardless of the underlying distribution of each individual bet.

This mathematical principle allows us to calculate precise probabilities for session outcomes. The distribution is characterized by two key parameters:

• Mean (μ): The expected value—your average outcome based on the house edge
• Standard Deviation (σ): The spread of outcomes—higher volatility means wider distribution

Variance: The Short-Term Wild Card

While the house edge determines your expected (average) outcome, variance determines how widely your actual results will spread around that average. As explained by researchers at the UNLV International Gaming Institute, variance is why gambling can feel exciting despite being mathematically unfavorable.

High variance games like slots or single-number roulette bets create dramatic swings—you might win big or lose big in any given session. Low variance games like even-money bets produce more consistent (though still typically negative) results. Our Variance Calculator explores this concept in more depth.

The Volatility Factor

Different bet types have vastly different volatility:

Why This Matters

Understanding outcome distributions helps with realistic expectations:

• Short sessions have wide distributions: 100 bets can result in significant wins or losses due to variance
• Longer sessions narrow toward expected value: Play 10,000 bets and your results will be very close to the mathematical expectation
• Win probability decreases over time: The more you play, the more certain your losses become
• Bankroll sizing matters: Understanding the range of outcomes helps set appropriate bankrolls

This is why our Time-to-Ruin Calculator shows that longer play always favors the casino. The distribution "tightens" around the expected negative value over time, converting short-term uncertainty into long-term mathematical certainty.

Statistical Concepts Explained

Confidence Intervals

A confidence interval tells you the range where a certain percentage of outcomes will fall. According to standard statistical practice:

• 68% confidence (1σ): About two-thirds of sessions will fall within one standard deviation of the mean
• 95% confidence (2σ): 19 out of 20 sessions will fall within two standard deviations
• 99.7% confidence (3σ): Virtually all sessions (997 out of 1000) will fall within three standard deviations

Z-Scores and Probability

The tool uses Z-scores—the number of standard deviations from the mean—to calculate exact probabilities. If your target outcome is 2 standard deviations above average, you can look up the probability in a standard normal distribution table. This is the same mathematics used in scientific research, quality control, and financial risk analysis.

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