Law of Large Numbers

#Definition

The Law of Large Numbers is a statistical theorem stating that as the number of trials increases, the average of observed outcomes converges to the expected value. In simpler terms: flip a fair coin enough times, and you'll get close to 50% heads, even if short sequences seem random or streaky.

In prediction markets, the Law of Large Numbers explains why market accuracy must be evaluated across many predictions, not single events. A market priced at 70% that resolves No isn't "wrong"—the 30% outcome simply occurred. Only by examining hundreds of 70% markets can we assess whether they actually resolve Yes about 70% of the time. This principle underlies all calibration analysis and justifies treating market prices as meaningful probability estimates.

#Why It Matters in Prediction Markets

The Law of Large Numbers is the mathematical foundation for trusting prediction market prices as probabilities.

Single predictions tell you nothing

When a 90% market resolves No, critics often claim the market "failed." The Law of Large Numbers explains why this criticism is misguided. A 10% event occurring once proves nothing; you need to observe many 90% markets to determine if they resolve Yes approximately 90% of the time.

Calibration becomes meaningful

The concept of calibration—whether markets' stated probabilities match actual frequencies—only makes sense because of the Law of Large Numbers. Checking if 60% predictions resolve Yes about 60% of the time requires enough predictions that short-term variance averages out.

Portfolio thinking over single bets

Traders who understand this law think in portfolios, not individual positions. A diversified set of positive expected value trades will converge toward expected profit over time, even if individual trades are unpredictable. This is why professional traders focus on edge across many trades rather than outcomes of single bets.

Distinguishes skill from luck

Over few trades, a lucky guesser looks like an expert. Over many trades, skill emerges as lucky and unlucky outcomes cancel out. The Law of Large Numbers is why track records need sufficient sample sizes to be meaningful.

#How It Works

#The Mathematical Statement

The Law of Large Numbers has two forms:

Weak Law: As sample size n increases, the probability that the sample average differs from the expected value by more than any small amount approaches zero.

Strong Law: As sample size approaches infinity, the sample average converges to the expected value with probability 1.

#Visualizing Convergence

#Python: Monte Carlo Convergence

Witness the Law of Large Numbers in action by simulating coin flips.

import random

def demo_convergence(total_flips=5000):
    heads = 0
    history = []
    
    for i in range(1, total_flips + 1):
        if random.random() < 0.5:
            heads += 1
        
        current_avg = heads / i
        history.append(current_avg)
        
        # Check specific checkpoints
        if i in [10, 100, 1000, 5000]:
            print(f"Flips: {i} | Heads %: {current_avg:.1%}")

demo_convergence()
# Output will show wild variance at 10, but tight convergence to 50% at 5000.

For a series of independent, identically distributed random variables
X₁, X₂, X₃, ... with expected value μ:

Sample average: X̄ₙ = (X₁ + X₂ + ... + Xₙ) / n

As n → ∞, X̄ₙ → μ (converges to expected value)

#Convergence in Practice

The law describes convergence, not immediate accuracy:

#Prediction Market Application

For a market priced at probability p:

After 1 resolution:
- Outcome is either 0 or 1
- No information about whether p is accurate

After 10 resolutions of similar markets:
- Observed frequency might be 40%-100% for p=70%
- Still too few to assess calibration

After 100 resolutions:
- Observed frequency likely within ±10% of p
- Starting to see if market is calibrated

After 1,000 resolutions:
- Observed frequency likely within ±3% of p
- Strong evidence about market accuracy

#Numerical Example: Calibration Assessment

A prediction market platform wants to assess if its markets are well-calibrated at the 80% probability level:

Data collection:
- Identified 500 markets that traded at $0.80 (80% implied)
- Tracked their resolutions
- 392 resolved Yes, 108 resolved No

Analysis:
- Observed frequency: 392/500 = 78.4%
- Expected frequency: 80%
- Difference: 1.6 percentage points

Interpretation:
- With 500 samples, standard error ≈ √(0.8×0.2/500) ≈ 1.8%
- Observed 78.4% is within normal variance of 80%
- Markets appear well-calibrated at this level

With only 20 samples:
- Might observe 14 Yes (70%) or 18 Yes (90%)
- Either would be within normal variance
- No meaningful conclusion possible

#The Gambler's Fallacy Connection

The Law of Large Numbers is often confused with the gambler's fallacy:

Correct: "After 1,000 flips, we'll be close to 50% heads"
Incorrect: "After 10 heads in a row, tails is more likely"

The convergence happens through dilution (more trials averaging out),
not through correction (increased probability of opposite outcomes).

#Sample Size Requirements

How many observations do you need to assess market accuracy?

For a given probability p and desired precision:

Standard error = √(p(1-p)/n)

To detect if true probability differs from stated by 5%:
- At p=50%: Need ~400 observations
- At p=80%: Need ~250 observations
- At p=95%: Need ~75 observations (but rare events are rare)

Practical rule of thumb:
- Minimum 50-100 observations per probability bucket
- 500+ for confident calibration assessment
- 1,000+ for detecting small miscalibrations

#Examples

#Example 1: Election Market Accuracy

A researcher evaluates prediction market accuracy on elections:

Single election:
- Market at 65%, candidate wins
- Is the market accurate? Impossible to say

20 elections:
- Markets averaging 65% (range 55-75%)
- 14 favorites won (70%)
- Some evidence of calibration, but noisy

200 elections:
- Markets at 60-70% range
- 129 favorites won (64.5%)
- Strong evidence of good calibration
- Deviation from 65% expected is within statistical noise

#Example 2: Sports Betting Calibration

A sports prediction market claims accurate probabilities:

Assessment by probability bucket:

Markets at 70-75%:
- 847 games, 612 favorites won (72.3%)
- Expected: ~72.5%
- Calibration: Excellent

Markets at 50-55%:
- 1,243 games, 678 favorites won (54.5%)
- Expected: ~52.5%
- Slight overconfidence, but within reason

Markets at 90-95%:
- 156 games, 147 favorites won (94.2%)
- Expected: ~92.5%
- Well-calibrated at extremes

#Example 3: Individual Trader Performance

A trader claims 65% win rate on prediction markets:

After 30 trades: 21 wins (70%)
- Consistent with 65% true rate? Yes
- Consistent with 50% true rate? Also yes (variance)
- Conclusion: Insufficient data

After 300 trades: 186 wins (62%)
- 95% confidence interval: ~57-67%
- Supports claimed 65% skill, but might be luck
- Verdict: Suggestive but not definitive

After 3,000 trades: 1,920 wins (64%)
- 95% confidence interval: ~62.3-65.7%
- Strong evidence of genuine ~64% skill
- Law of Large Numbers has revealed true ability

#Example 4: Resolution Variance

A prediction market prices an event at $0.30 (30% probability):

Single market resolution:
- If No: "See, markets are great!" (but 70% of time)
- If Yes: "Markets are unreliable!" (but 30% is expected)
- Neither conclusion is valid from one observation

Across 100 similar markets priced at 30%:
- Expected Yes resolutions: ~30
- Actual Yes resolutions: 28
- Deviation is normal; markets appear calibrated

The Law of Large Numbers explains why we need 100 markets,
not 1, to assess if 30% probabilities are accurate.

#Risks and Common Mistakes

Drawing conclusions from small samples

The most common error: evaluating prediction market accuracy from a handful of events. Claiming markets "failed" because one 80% prediction was wrong, or "succeeded" because one 60% prediction was right, ignores the fundamental statistics of probability.

Expecting convergence in short timeframes

The law describes asymptotic behavior—convergence as samples approach infinity. In finite samples, substantial deviation is normal. A 50/50 prediction might go 7-3 over 10 trials without being "wrong." Patience is required.

Confusing individual and aggregate accuracy

An individual market being wrong tells you nothing. Aggregate patterns across many markets tell you everything. Traders who evaluate market reliability by cherry-picking individual outcomes misunderstand the statistical basis for prediction markets.

Ignoring independence requirements

The Law of Large Numbers assumes independent trials. If prediction errors are correlated (all political markets are wrong due to a polling bias), the law doesn't apply as simply. Diversifying across truly independent events matters.

Using small samples for calibration claims

Claiming your trading system is "80% accurate" after 20 trades is statistically meaningless. The confidence interval around 80% with n=20 is roughly 56-94%. You might actually be a 60% or 95% trader.

#Practical Tips for Traders

• Think in portfolios, not individual bets: The Law of Large Numbers only helps you if you make many trades. A single large bet has high variance regardless of your edge. Diversification across many independent markets allows the law to work in your favor

• Require sufficient sample sizes for track record evaluation: Before trusting your own (or anyone's) prediction accuracy, demand enough observations for statistical significance. 30 trades is a start; 100 is better; 300+ allows confident assessment

• Don't update beliefs strongly on single outcomes: One market resolving unexpectedly shouldn't dramatically change your view of prediction market reliability. Even a 95% probability has a 1-in-20 chance of being wrong

• Understand variance in your expected results: If you have 60% edge, expect to lose 40% of bets. A streak of 5 losses among 10 bets is within normal variance. Don't abandon a sound strategy due to short-term results

• Use the law to set realistic expectations: With 100 trades at 55% win rate, expect roughly 55 wins. But also expect actual results to range from maybe 45-65 wins. Plan bankroll management for this variance

• Separate calibration assessment from outcome observation: To evaluate if markets (or your estimates) are calibrated, track predictions by probability bucket over many events. This requires record-keeping and patience

• Remember: the law is about averages, not guarantees: Even over many trials, you're only guaranteed to be close to expected value, not exactly at it. There's always residual variance, just less of it

#Related Terms

• Calibration
• Expected Value (EV)
• Bayes' Theorem
• Efficient Market Hypothesis
• Gambler's Fallacy
• Risk Management
• Kelly Criterion

#FAQ

#How many predictions are needed to evaluate market accuracy?

It depends on the precision you need and the probability levels you're assessing. For rough calibration assessment, 50-100 observations per probability bucket is a minimum. For confident conclusions about small miscalibrations, 500-1,000+ observations are needed. Single predictions or even dozens are insufficient to draw meaningful conclusions about whether market prices accurately reflect probabilities.

#Does the Law of Large Numbers mean prediction markets are always right?

No. The law says that if market prices are accurate probabilities, then over many markets, resolution frequencies will match those probabilities. It doesn't guarantee accuracy—it provides the framework for testing accuracy. Markets could be systematically miscalibrated, which would show up as consistent deviation between stated probabilities and resolution frequencies across large samples.

#How does this relate to the gambler's fallacy?

They're often confused. The Law of Large Numbers says frequencies converge to probabilities over many trials. The gambler's fallacy incorrectly believes individual trials must "balance out" past results. The convergence happens through dilution (many future trials overwhelming past deviations), not correction (increased probability of opposite outcomes). Each trial remains independent regardless of past results.

#Can I use the Law of Large Numbers to guarantee profits?

Not directly. If you have positive expected value across many trades, the law suggests your average outcome will converge to that positive expectation. But "many" might mean hundreds or thousands of trades, and variance along the way can be substantial. You need sufficient bankroll to survive losing streaks, and your edge estimate must actually be correct. The law doesn't create edge; it just describes how edge manifests over time.

#Why do prediction market critics cite individual wrong predictions?

Usually misunderstanding of probability or motivated reasoning. Someone who wants to discredit prediction markets can always find an individual market that resolved against the probability. But this proves nothing—a 90% probability being wrong once is expected. The valid criticism would be: "Across 1,000 markets priced at 90%, only 75% resolved as predicted." That would indicate miscalibration. Cherry-picking single outcomes is statistically invalid.

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Meta Description (150-160 characters): Learn the Law of Large Numbers and why prediction market accuracy must be measured across many events, not single predictions. Essential for calibration analysis.

Secondary Keywords Used:

• statistical convergence
• probability convergence
• sample size
• long-run accuracy
• calibration