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Mining Luck & Variance

Understanding the probabilistic nature of block discovery

Contents

Mining as a Poisson Process

Bitcoin mining is fundamentally a Poisson process. Each hash has an independent, identically distributed (i.i.d.) probability of finding a valid block. This means:

  • Block discovery follows an exponential distribution
  • Past results don't affect future probabilities ("memoryless")
  • Multiple block discoveries follow a Poisson distribution
  • Variance is inherent and unavoidable

Why This Matters

Because mining is probabilistic, pools don't find blocks at a constant rate. A pool with 10% hashrate might find 8 blocks one day and 15 the next — both are statistically normal outcomes.

Poisson Probability $$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$
Where
$X$
Number of blocks found
$k$
Specific number of blocks
$\lambda$
Expected blocks (mean of distribution)
$e$
Euler's number (≈ 2.718)

Luck Calculation

"Luck" quantifies how a pool's actual block discovery compares to statistical expectation:

Mining Luck $$\text{Luck} = \frac{\text{Actual Blocks}}{\text{Expected Blocks}} \times 100\%$$
< 90%
Unlucky
90-95%
Below Average
95-105%
Normal
105-110%
Above Average
> 110%
Lucky

Interpretation

Luck % Meaning Example (Expected: 100 blocks)
100% Exactly as expected Found exactly 100 blocks
120% 20% more than expected (lucky) Found 120 blocks
80% 20% less than expected (unlucky) Found 80 blocks
Long-Term Convergence

Over long periods, luck converges toward 100%. Short-term deviations are normal and expected. The larger the pool (more expected blocks), the faster convergence occurs.

Expected Blocks

Expected blocks are calculated from the pool's hashrate share:

Expected Blocks $$E[\text{Blocks}] = \text{Pool Share} \times \text{Total Blocks}$$

ELEKTRON's Approach

Since we can't directly measure pool hashrate, we estimate share from recent block history using a rolling window:

Rolling Share Estimation $$\text{Share}_{t} = \frac{\sum_{i=t-w}^{t} B_{pool,i}}{\sum_{i=t-w}^{t} B_{total,i}}$$
Where
$w$
Rolling window (default: 30 days)
$B_{pool,i}$
Blocks found by pool on day $i$
$B_{total,i}$
Total blocks on day $i$

Using rolling share (instead of fixed share) accounts for pools growing or shrinking over time, giving a more accurate luck calculation.

Variance & Convergence

Poisson Variance

For a Poisson distribution, variance equals the mean:

Poisson Properties $$\text{Var}(X) = \lambda = E[X]$$

This has important implications:

Standard Deviation of Luck

Luck Standard Deviation $$\sigma_{\text{luck}} = \frac{1}{\sqrt{\lambda}} \times 100\%$$
Where
$\lambda$
Expected blocks in the period

Variance vs Pool Size

Expected Blocks/Day ~Pool Share Luck Std Dev 95% Range
1 0.7% ±100% 0-300%
4 2.8% ±50% 0-200%
14 10% ±27% 46-154%
43 30% ±15% 70-130%
Small Pool Variance

Small pools experience extreme luck swings. A 1% pool might find 0 blocks one day (0% luck) and 4 blocks the next (400% luck) — both within normal statistical bounds.

Rolling Luck Analysis

To smooth out daily noise, ELEKTRON calculates rolling luck over windows:

Rolling Luck (7-Day) $$\text{Luck}_{7d,t} = \frac{\sum_{i=t-6}^{t} B_{actual,i}}{\sum_{i=t-6}^{t} B_{expected,i}} \times 100\%$$

Window Selection

Window Use Case Trade-off
1 day Real-time monitoring High noise, immediate signal
7 days Weekly performance Balanced noise/signal
30 days Monthly assessment Low noise, lagged signal
90 days Long-term trend Very smooth, misses short-term

Coefficient of Variation

The coefficient of variation (CV) normalizes variance relative to the mean, allowing comparison across pools of different sizes:

Coefficient of Variation $$CV = \frac{\sigma}{\mu} \times 100\%$$
Where
$\sigma$
Standard deviation of daily luck
$\mu$
Mean luck (should be ~100% over time)

CV Interpretation

CV Range Interpretation
< 15% Very stable (large pool)
15-30% Moderate variance
30-50% High variance (small pool)
> 50% Extreme variance (very small pool)

Dominance-Luck Correlation

ELEKTRON tracks the relationship between pool size (dominance) and luck variance. Larger pools have lower variance due to the law of large numbers. This creates "convergence thresholds" — hashrate shares where luck stabilizes within certain ranges.

Practical Insight

A pool needs approximately 5-10% network share for daily luck to consistently stay within ±20% of expected. Smaller pools should evaluate luck over longer periods (weeks/months) to avoid noise-driven conclusions.