Joint Probability

#Definition

Joint probability is the likelihood that two or more events will both occur. In prediction markets, joint probability measures the chance that multiple outcomes happen together—such as a candidate winning both a primary and general election, or two related economic indicators both exceeding expectations.

Expressed as P(A and B), joint probability is fundamental to evaluating multi-leg positions, understanding correlated markets, and avoiding overexposure to linked outcomes.

#Why It Matters in Prediction Markets

Joint probability governs how traders should think about portfolios of related positions:

Correlated risk assessment: Holding positions in related markets means your outcomes are linked. Joint probability quantifies how often you'll win (or lose) on multiple positions simultaneously.

Multi-leg strategy evaluation: Strategies that require multiple events to occur (e.g., "Candidate X wins AND passes Policy Y") depend on joint probability for accurate expected value calculations.

Diversification reality check: Apparent diversification may be illusory if positions share joint probability. Ten "different" markets that all depend on the same underlying factor aren't truly diversified.

Arbitrage detection: When market prices imply joint probabilities that violate mathematical constraints, arbitrage opportunities exist.

Portfolio drawdown estimation: The probability of simultaneous losses across positions—a joint probability—determines worst-case drawdown scenarios.

#How It Works

#Basic Formula

For two events A and B:

P(A and B) = P(A) × P(B|A)

Or equivalently:

P(A and B) = P(B) × P(A|B)

Where P(B|A) is the conditional probability of B given A.

#Visualizing the Intersection

P(A and B) is where outcomes overlap.

#Python: Calculating Joint Probability

Tool to calculate joint probability assuming independence vs. conditional dependence.

def calc_joint_prob(p_a, p_b, p_b_given_a=None):
    """
    Calculates P(A and B).
    If p_b_given_a is None, assumes Independence.
    """
    if p_b_given_a is None:
        # P(A) * P(B) -> Only if independent!
        joint = p_a * p_b
        return joint, "Assumed Independent"
    else:
        # P(A) * P(B|A) -> General rule
        joint = p_a * p_b_given_a
        return joint, "Calculated with Dependence"

# Example: 
# P(Primary Win) = 0.70
# P(General Win | Primary Win) = 0.60
p, method = calc_joint_prob(0.70, 0.60, 0.60) # Note: p_b is not used if conditional provided
print(f"Joint Probability: {p:.1%} ({method})")

# Example if mistakenly assumed independent (P(General) approx 0.40)
p_wrong, method = calc_joint_prob(0.70, 0.40) 
print(f"Naive Indep. Calc: {p_wrong:.1%} ({method})")

#Independent Events

When events are independent (one doesn't affect the other):

P(A and B) = P(A) × P(B)

Example: A coin flip and a die roll are independent.

• P(Heads) = 0.50
• P(Roll 6) = 0.167
• P(Heads AND Roll 6) = 0.50 × 0.167 = 0.083

#Dependent Events in Prediction Markets

Most prediction market events are NOT independent. A candidate's primary win affects their general election odds. An economic indicator affects related policy decisions.

Example with dependent events:

• P(Candidate wins primary) = 0.70
• P(Candidate wins general | wins primary) = 0.60
• P(Candidate wins general | loses primary) = 0 (can't win general without primary)

Joint probability of winning both:

P(Primary AND General) = P(Primary) × P(General | Primary)
P(Primary AND General) = 0.70 × 0.60 = 0.42

The candidate has a 42% chance of winning both races.

#Three or More Events

For multiple events:

P(A and B and C) = P(A) × P(B|A) × P(C|A and B)

Example: Probability a candidate wins primary, general, AND passes signature legislation:

• P(Primary) = 0.70
• P(General | Primary) = 0.60
• P(Legislation | Primary and General) = 0.40

P(All three) = 0.70 × 0.60 × 0.40 = 0.168

Only 16.8% chance all three occur—much lower than any individual probability.

#Extracting Joint Probability from Market Prices

When separate markets exist for related events, you can infer joint probabilities:

Market A: "Candidate wins primary" at $0.75 **Market B**: "Candidate wins general" at$0.50

If winning the general requires winning the primary:

P(General) = P(Primary) × P(General | Primary)
0.50 = 0.75 × P(General | Primary)
P(General | Primary) = 0.667

The joint probability P(Primary AND General) equals P(General) = 0.50 since winning the general implies winning the primary.

#Examples

Sequential political events: A trader evaluates positions in "Party X wins House" ($0.55) and "Party X wins Senate" ($0.45). These aren't independent—national mood affects both. Historical correlation suggests P(Both | House) ≈ 0.70. Joint probability of winning both: 0.55 × 0.70 = 0.385. A market pricing "Party X wins both chambers" above $0.385 may be overpriced.

Economic indicator chain: Markets exist for "Inflation exceeds 4%" ($0.30) and "Fed raises rates" ($0.60). The Fed decision is conditional on inflation. If P(Rate hike | High inflation) = 0.90 and P(Rate hike | Low inflation) = 0.45:

P(High inflation AND Rate hike) = 0.30 × 0.90 = 0.27

A trader holding both Yes positions has 27% joint probability of winning both.

Sports parlay equivalent: A trader bets on three binary sports outcomes, each at $0.60 (60% probability). If outcomes are independent:

P(All three win) = 0.60 × 0.60 × 0.60 = 0.216

Winning all three happens only 21.6% of the time despite each individual bet having 60% odds.

Correlated portfolio risk: A trader holds five election market positions, each with 55% win probability. If independent, P(All five lose) = 0.45^5 = 0.018 (1.8%). But elections are correlated. If actual P(All five lose) = 0.10 due to common factors, the portfolio is riskier than independent analysis suggests.

#Risks and Common Mistakes

Assuming independence when events are correlated: The most common error. Traders calculate joint probability as P(A) × P(B) when events share underlying factors. This underestimates both upside correlation (winning together) and downside correlation (losing together).

Ignoring conditional structure: Joint probability requires conditional reasoning. P(A and B) isn't just about A and B in isolation—it requires understanding how A affects B.

Multiplication without thought: Multiplying many probabilities together produces small numbers that feel "unlikely." But if you're betting on many outcomes, rare joint events happen regularly across your portfolio.

Probability greater than components: Joint probability P(A and B) cannot exceed P(A) or P(B). If market prices imply otherwise, someone is wrong—or an arbitrage exists.

Confusing joint and conditional: P(A and B) is not the same as P(A|B). Joint probability is absolute; conditional probability is relative to a condition being true. Mixing these up produces incorrect calculations.

Neglecting impossible combinations: Some joint events are impossible by definition (a candidate can't win AND lose the same race). Joint probability of mutually exclusive events is zero.

#Practical Tips for Traders

• Map dependencies before calculating: Identify which events affect which others. Draw the causal/logical structure before applying formulas.

• Use conditional markets when available: Some platforms offer conditional markets that directly price P(A|B). These simplify joint probability calculation.

• Stress test for correlation: When estimating joint probability of losses, assume higher correlation than seems obvious. Hidden common factors often emerge during stress.

• Check for arbitrage: If calculated joint probabilities from separate markets violate mathematical constraints (negative probabilities, probabilities > 1), trading opportunities exist.

• Size portfolios to joint risk: Your worst-case isn't losing one position—it's losing correlated positions together. Size based on joint probability of multiple losses, not individual position risk.

• Track correlation empirically: Over time, record how often your positions win/lose together. Compare to your assumed joint probabilities to calibrate.

• Remember multiplication shrinks fast: Three 70% events have joint probability 34%. Five 70% events: 17%. Compound positions require much higher individual probabilities than intuition suggests.

#Related Terms

• Conditional Probability
• Expected Value (EV)
• Arbitrage
• Drawdown
• Risk Management
• Correlation
• Kelly Criterion
• Information Aggregation

#FAQ

#What is joint probability in simple terms?

Joint probability is the chance that two or more things both happen. In prediction markets, it answers questions like "What's the probability that Candidate X wins the primary AND wins the general election?" It's calculated by multiplying probabilities together, accounting for how events affect each other.

#How does joint probability differ from conditional probability?

Conditional probability P(A|B) is the probability of A given that B is true—it assumes B happened. Joint probability P(A and B) is the probability that both A and B occur—neither is assumed. They're related: P(A and B) = P(A|B) × P(B). Joint is absolute; conditional is relative to a condition.

#Why can't I just multiply individual market prices together?

You can only multiply directly if events are independent—meaning one doesn't affect the other. Most prediction market events share underlying factors (national mood, economic conditions, common causes). Multiplying without accounting for dependence gives wrong answers, usually underestimating correlation.

#What happens to joint probability with many events?

It shrinks rapidly. Even high-probability events, when combined, produce low joint probability. Five events each at 80% probability have joint probability of 0.80^5 = 32.8% if independent. This is why multi-leg bets are risky and why "parlay" strategies often lose despite individual legs having positive expected value.

#How can joint probability create arbitrage opportunities?

Mathematical constraints require joint probabilities to be consistent. If P(A) = 0.60, P(B) = 0.50, and P(A and B) implied by market prices is 0.55, something is wrong—joint probability can't exceed either individual probability. When markets violate these constraints, traders can construct positions that profit regardless of outcomes.