Probability Distribution

#Definition

A probability distribution is a mathematical function that describes the likelihood of all possible outcomes for a random variable. In prediction markets, probability distributions reveal not just the expected outcome but the full range of possibilities and their relative likelihoods.

While a single market price gives a point estimate (e.g., 60% probability), understanding the underlying distribution reveals richer information—the shape of uncertainty, the likelihood of extreme outcomes, and the confidence around that estimate.

#Why It Matters in Prediction Markets

Probability distributions provide essential context beyond point estimates:

Understanding uncertainty range: A 60% probability could mean "fairly confident" or "highly uncertain with a slight lean." The distribution tells you which.

Portfolio risk assessment: The distribution of portfolio outcomes—not just the expected return—determines risk of drawdown and likelihood of various profit/loss scenarios.

Tail risk identification: Distributions reveal the probability of extreme outcomes that point estimates hide. A "small" 5% probability event happens 1 in 20 times.

Strategy evaluation: Using Monte Carlo simulation to generate outcome distributions helps test strategies against the full range of possible futures.

Multiple outcome markets: Markets with more than two outcomes (e.g., "Which candidate wins?") inherently involve distributions across all possibilities.

#How It Works

#Basic Concepts

Discrete distributions: For outcomes that can only take specific values (binary markets: Yes/No; categorical: Candidate A/B/C/D).

Continuous distributions: For outcomes that can take any value in a range (e.g., vote share percentage, economic indicators).

Key properties:

• All probabilities sum to 1 (or integrate to 1 for continuous)
• Each outcome has a probability between 0 and 1
• The distribution captures the full range of uncertainty

#Binary Markets

The simplest distribution: two outcomes with probabilities p and (1-p).

Example:

• Market price: $0.60 (60% Yes probability)
• Distribution: P(Yes) = 0.60, P(No) = 0.40

Even binary markets involve distribution thinking when considering expected value:

• If you buy Yes at $0.60 and win, you gain$0.40
• If you lose, you lose $0.60
• EV = (0.60 × $0.40) - (0.40 ×$0.60) = $0.24 -$0.24 = $0

#Multi-Outcome Markets

Markets with multiple exclusive outcomes have discrete distributions.

Example: Primary election with 5 candidates

This distribution shows not just who's favored but the relative likelihood of each outcome.

#Portfolio Outcome Distributions

When holding multiple positions, the portfolio return has its own distribution.

Example: Three independent positions, each with 60% win probability and $100 stake

Possible outcomes:

• All 3 win: 0.6³ = 21.6% probability, +$300
• 2 win, 1 lose: 3 × 0.6² × 0.4 = 43.2% probability, +$100
• 1 win, 2 lose: 3 × 0.6 × 0.4² = 28.8% probability, -$100
• All 3 lose: 0.4³ = 6.4% probability, -$300

The distribution of outcomes ranges from -$300 to +$300, with +$100 being most likely.

#Common Distribution Shapes

Symmetric/Normal: Outcomes equally likely above and below the mean. Bell-shaped curve.

Skewed right: Long tail of positive outcomes (potential for large gains, limited losses).

Skewed left: Long tail of negative outcomes (potential for large losses, limited gains).

Fat-tailed: Extreme outcomes more likely than normal distribution suggests. Common in financial markets.

Bimodal: Two distinct peaks, suggesting two likely scenarios with less probability in between.

#Numerical Example: Vote Share Distribution

A prediction market implies a candidate will receive 52% of the vote. But what does the distribution look like?

Tight distribution (high confidence):

• 68% chance of 50-54%
• 95% chance of 48-56%
• Very unlikely below 45% or above 60%

Wide distribution (high uncertainty):

• 68% chance of 45-59%
• 95% chance of 38-66%
• Meaningful probability of landslide in either direction

Same point estimate (52%), very different distributions and trading implications.

#Examples

Election seat count: A market shows Party A winning an average of 52 Senate seats. The distribution matters: is it 50-54 with 90% probability, or 45-60? If your trading strategy depends on controlling the Senate (50+ seats), you need the distribution, not just the average.

Portfolio drawdown analysis: You hold 10 prediction market positions. Monte Carlo simulation generates the distribution of possible portfolio outcomes: 50% chance of +15% or better, but 10% chance of -25% or worse. The distribution reveals tail risk that expected value alone doesn't show.

Conditional outcome distributions: If Candidate X wins the primary, what's the distribution of general election outcomes? This conditional distribution might be bimodal: either they win convincingly (strong candidate) or lose badly (polarizing figure), with fewer outcomes in between.

Event timing markets: Markets on "when" something happens have continuous distributions. The distribution might show highest probability in months 3-6, declining probability thereafter, with a long tail extending years out.

#Risks and Common Mistakes

Ignoring the distribution: Focusing only on expected value or market price without considering the range of outcomes. A trade with positive EV can still have high probability of loss.

Assuming normal distributions: Financial and prediction market outcomes often have fat tails—extreme events happen more often than normal distributions predict. Underestimating tail risk is dangerous.

Misinterpreting point estimates: A 60% probability doesn't mean "it will probably happen." It means 40% of the time it won't. Over many such events, 40% outcomes occur regularly.

Ignoring correlation effects: When positions are correlated, the portfolio distribution is more extreme than independent analysis suggests. Both upside and downside tails become fatter.

Overconfidence in distribution estimates: The true distribution is unknown. Estimated distributions reflect assumptions that may be wrong. Treat distribution analysis as informative, not definitive.

#Practical Tips for Traders

• Think in distributions, not points: When evaluating a position, consider the range of outcomes and their probabilities, not just the expected value.

• Use Monte Carlo for portfolios: Simulate many possible outcomes to understand your portfolio's true distribution of returns, especially tail risks.

• Examine the tails: Pay special attention to extreme outcomes. Low probability events with high impact matter for risk management.

• Consider distribution shape: Skewed distributions mean median and mean differ. Understand which measure is relevant for your decisions.

• Track calibration across outcomes: If your 70% predictions come true 70% of the time, your probability distributions are well-calibrated. If not, adjust.

• Size for worst-case scenarios: Position sizing should account for adverse outcomes in the distribution, not just expected outcomes.

• Update distributions with new information: As information arrives, the distribution narrows (or shifts). Bayesian updating refines your probability estimates.

#Related Terms

• Expected Value (EV)
• Volatility
• Monte Carlo Simulation
• Conditional Probability
• Risk Management
• Drawdown
• Kelly Criterion
• Correlation

#FAQ

#What is a probability distribution in simple terms?

A probability distribution shows all possible outcomes and how likely each one is. Instead of saying "60% chance of winning," it says "60% chance of winning, 40% chance of losing, and if you win, here's how much you might win by." It's the complete picture of uncertainty rather than a single number.

#How do probability distributions relate to prediction market prices?

Prediction market prices are point estimates from the underlying distribution. A binary market price of $0.65 implies a distribution: 65% probability of Yes, 35% probability of No. Multi-outcome markets show distributions directly across candidates or outcomes. The price is a summary; the distribution is the full picture.

#Why do fat tails matter in prediction markets?

Fat tails mean extreme outcomes happen more often than "normal" statistics suggest. In prediction markets, this means big upsets, major surprises, and black swan events are more common than naive probability estimates predict. Ignoring fat tails leads to underestimating risk and overestimating the predictability of outcomes.

#How can I estimate the distribution behind a market price?

For multi-outcome markets, prices directly show the distribution. For binary markets, the distribution is simple (p and 1-p). For continuous outcomes, you can: (1) look at historical volatility of similar events, (2) examine options-like instruments if available, (3) use Monte Carlo simulation with reasonable assumptions, or (4) analyze the dispersion of expert forecasts as a proxy.

#What's the difference between expected value and distribution?

Expected value is a single number—the probability-weighted average outcome. The distribution shows all possible outcomes and their probabilities. You can have the same expected value with very different distributions: one might be concentrated around the mean, another might be spread wide with fat tails. Distribution captures risk; expected value captures average outcome.