Mining Variance & Poisson Math

A miner sets up a single Antminer S21+ pointed at Bitcoin Cash. The math says the expected time to find a block is ~133 days. The miner waits 130 days and finds nothing. They wait 140 days and still nothing. They check the dashboard at 150, 160, 180 days — still no block. Has the math failed? Is the rig broken? Did they pick the wrong pool?

The answer, almost always, is none of the above. The math hasn't failed. The rig is fine. The pool is fine. The miner is experiencing variance — the gap between long-run expected value and short-term reality. And in solo mining, that gap can be enormous. 1 in 5 single-rig miners will wait more than 1.5× their mean time before finding a block. 1 in 20 will wait more than 3×. These aren't rare outliers. They're predicted by the math.

This article exists because most solo mining advice handwaves the variance problem. "Mean time is X days" sounds simple, like a deterministic countdown. It isn't. Solo mining is a Poisson process — memoryless, exponential, prone to clustered outcomes — and understanding the math is the difference between thinking your hardware is broken and recognizing that you're inside a perfectly normal slow tail of the probability distribution.

This is the gufo's mathematical guide to solo mining. We'll walk through Poisson basics, the exponential distribution that governs time-to-block, real percentile tables for common hardware, Monte Carlo simulations, and a clear framework for emotional preparation. By the end, "long dry stretches" will look like math, not failure.

The setup: why mining is a Poisson process

Mining hashes one number at a time, looking for a value below the network target. Each hash is statistically independent — the SHA-256 algorithm makes the previous hash's outcome irrelevant to the next attempt. This is the textbook setup for a Poisson process.

A Poisson process has three defining properties:

1. Events occur independently. Each hash is a fresh attempt; finding (or not finding) a block doesn't change the probability of the next attempt.
2. Events occur at a constant average rate. Network difficulty stays roughly stable on short timescales (days). Your hashrate stays stable. So the expected rate is constant.
3. The probability of an event in any tiny time interval is proportional to its length. Twice as long mining, twice as much probability of finding a block.

All three are satisfied for solo mining. The math that applies:

• The number of blocks found in time period T follows a Poisson distribution with parameter λ = (your_hashrate ÷ network_hashrate) × (T ÷ 600 seconds)
• The time between consecutive blocks follows an exponential distribution with mean = 1/λ
• The probability of finding zero blocks in time T is e^(-λT)
• The standard deviation of expected blocks in time T equals √(λT) — the square root of the mean

These three formulas are the entire mathematical foundation of solo mining variance. Memorize them and the rest is application.

The mean: what it actually means

"Mean time to find a block" is the most cited number in solo mining and the most misunderstood. The mean is NOT what you should expect to wait. It's the long-run average across many trials. Single trials can produce wildly different outcomes.

For an exponential distribution (which governs time-between-blocks):

Median time = 0.693 × Mean time

That is, half of all single-trial outcomes will take less than 69.3% of the mean time. The other half will take longer. The distribution is right-skewed — there's a long tail of unlucky outcomes that drag the mean upward.

For a single Antminer S21+ on BCH (mean ~133 days):

• ~50% chance of finding a block by day 92 (median)
• ~63% chance by day 133 (mean)
• ~80% chance by day 215
• ~90% chance by day 306
• ~95% chance by day 399
• ~99% chance by day 612
• ~1% chance you'll still be waiting after 612 days

Read the last bullet again. Even with a "133-day mean," there's a 1 in 100 chance you'll go more than 600 days without a block. That's not 1 in a million — that's 1 in 100. If 1,000 miners all run identical S21+ setups on BCH for two years, statistically about 10 of them will go the full two years with zero blocks found. Not because of bad luck or broken hardware — because of the math.

The exponential distribution, visualized

The probability of finding a block by time t (cumulative distribution function) is:

P(find by time t) = 1 - e^(-t/μ)

where μ is the mean time. For a 133-day mean, the percentile breakdown:

Note the key insight: even at 4× mean (532 days for a 133-day mean), there's still a 1.8% chance of zero blocks. The exponential distribution has a fat right tail. Long dry stretches are mathematically guaranteed for some fraction of miners.

The variance and standard deviation

For Poisson processes, the variance equals the mean. The standard deviation is √mean. This has practical implications:

Over a year (365 days) of mining a single S21+ on BCH:

• Expected blocks: 365 ÷ 133 = ~2.74 blocks
• Standard deviation: √2.74 = 1.66 blocks
• ~68% confidence interval: 1.08 to 4.41 blocks
• ~95% confidence interval: 0 to 6.06 blocks
• ~99% confidence interval: 0 to 7.72 blocks

So in a typical year you might find 1, 2, 3, 4, or 5 blocks — all within normal statistical range. Finding 0 blocks (which is in the 95% range tail) is unusual but not extreme. Finding 7+ blocks is also unusual but not extreme. The actual single-year outcome can range from 0 to 7+ blocks while still being entirely consistent with the math.

For the SoloFury fleet (4× S21+, ~940 TH/s on BCH), expected blocks per year:

• Mean time per block: 133 ÷ 4 = ~33 days
• Expected blocks/year: 365 ÷ 33 = ~11 blocks
• Standard deviation: √11 = 3.3 blocks
• ~95% range: 4.4 to 17.6 blocks/year

SoloFury found 3 BCH blocks in the first 19 days of late April / early May 2026. That's an annualized rate of ~58 blocks/year — way above the ~11 expected. That's a positive variance excursion of ~13 standard deviations above the mean for that specific window, which is genuinely unusual but possible. Equivalently: there were probably some 19-day windows in 2025 where the same fleet found zero blocks, and those wouldn't have been newsworthy because they look "normal."

The clustering problem

Poisson processes have a counterintuitive property: events tend to cluster rather than space evenly. If you find a block today, there's no extra "unluckiness" that comes after — your probability of finding another block tomorrow is exactly the same as it was before. Over short windows, this leads to runs of multiple blocks close together, separated by long dry periods.

For example, the SoloFury BCH blocks #947633 (April 20), #948592 (April 27), and #950338 (May 9) — three blocks in 19 days. Then nothing for a stretch (typical pattern). Then maybe two more in a week. Then nothing for two months.

The pattern isn't broken — it's exactly what Poisson predicts. Block-finding is "memoryless": the network doesn't remember that you just found a block, doesn't punish you for being lucky, doesn't reward you for being patient. Each new attempt is a fresh roll of the dice.

The clustering is what makes solo mining feel emotionally chaotic. Months of nothing followed by abrupt jackpots, then more nothing. If you map the actual block timing, you'll see clusters and gaps in roughly equal measure. Both are normal.

Monte Carlo simulation: 10,000 simulated years

Numbers are abstract. Let's run a simulation. Imagine 10,000 identical solo miners running an S21+ on BCH for 365 days each (so 10,000 simulated mining years). What does the distribution of yearly outcomes look like?

Read this carefully:

• ~6.5% of single-rig miners will find ZERO blocks in any given year. Not because they're doing anything wrong. Because variance.
• ~30% will find more than the mean (3+ blocks) — luck working in their favor.
• ~3% will have a "jackpot year" with 7+ blocks — extremely lucky, but mathematically expected for some fraction.
• The total revenue across all 10,000 miners averages out to the expected value (~2.74 blocks each), but individual experiences vary enormously.

Some miners will have 3 great years in a row and conclude "I have the magic touch." Others will have 2 zero-block years in a row and conclude "solo mining doesn't work." Both are reading too much into too few data points. The math says: run the experiment longer, and individual outcomes converge to the mean.

The "Gambler's Fallacy" trap

Many solo miners fall into a classical reasoning error: "I haven't found a block in 200 days, so I'm 'due' for one." This is wrong. Poisson processes are memoryless. The probability of finding a block in the next 30 days, given you've gone 200 days without one, is exactly the same as the probability of finding one in your first 30 days.

The math: P(find in next 30 days | gone 200 days without one) = P(find in any 30-day window) = 1 - e^(-30/133) = 20.2%

The 200 days of dry stretch don't help. They don't hurt either. They just don't matter. Dice have no memory.

The reverse is also true: just because you found a block last week doesn't mean you're "less likely" to find one this week. Your probability is unchanged. Lucky streaks don't get punished. Unlucky streaks don't get compensated. Each new attempt is independent.

The miners who internalize this are the ones who maintain steady operation over long timeframes. The miners who don't get emotional, change strategies during bad stretches, give up at exactly the wrong moment, and miss the jackpot when it eventually arrives.

The variance scaling: more rigs means less relative variance

Here's where the math gets practically useful: variance scales with the square root of the mean, but expected revenue scales linearly with the mean.

So if you double your hashrate:

• Expected blocks per year: 2× (twice as many)
• Standard deviation of blocks: √2 ≈ 1.41× (only 41% more spread)
• Coefficient of variation (std dev / mean): drops by 1/√2 ≈ 29%

In other words, larger fleets experience proportionally less variance. The big farms aren't lucky — they're mathematically smoothed by their scale.

For a single rig, year-over-year revenue can swing 60%+. For a 100-rig farm, swings are typically under 6%. This is why industrial mining is a lower-variance business than solo mining — not because the math changes, but because scale averages out the noise.

Variance vs gambling: the structural difference

People often equate solo mining with gambling because both involve probability, both have negative-expected-value scenarios at small scales, and both have winners and losers. Mathematically, they're different in important ways:

Gambling (e.g., Powerball, casino)

• House edge: the math is structured to favor the operator over the long run
• Negative expected value by design — players lose on average
• Independent attempts but rigged probabilities — your individual hashes don't matter, just whether your specific ticket matches

Solo mining

• No house edge: the protocol issues block rewards based on cryptographic work; no one is taking a percentage off the top (except your 1% pool fee, which is operational not structural)
• Positive or near-zero expected value depending on hardware and electricity costs — at SoloFury's 1% fee structure, expected value is essentially "your hardware capability minus your costs"
• Independent attempts, fair probabilities — your hashes contribute to the cumulative network attempt, your share is proportional to your hashrate

The variance is real. The structural unfairness isn't. Solo mining is high-variance honest work. Gambling is low-variance dishonest work. The math is similar; the structures couldn't be more different.

Real-world percentile tables

For miners considering solo mining, here are the percentiles for different hardware setups on BCH:

Single Bitaxe Gamma (1.2 TH/s) on BC2 (mean ~1.7 days)

Bottom line: at 1.2 TH/s on BC2, you'll almost certainly find at least one block within 5 days. ~5% chance of waiting longer. ~0.3% chance of going more than 10 days. Practically, this is an exciting "active" mining experience.

Single Antminer S21+ (235 TH/s) on BCH (mean ~133 days)

Bottom line: in 1 year of operation, ~93.5% chance of at least one block. ~6.5% chance of zero blocks. The 6.5% miners aren't doing anything wrong; they're just at the bad end of the distribution.

Single Bitaxe Gamma (1.2 TH/s) on BTC (mean ~12,000 years)

Bottom line: lottery mode. The probability is non-zero but vanishingly small over any human timescale. Some Bitaxe operators have hit anyway because variance combined with thousands of operators globally produces occasional jackpots. The cumulative probability across all Bitaxe owners is meaningfully higher than any individual's probability.

The emotional preparation framework

Now that the math is on the table, here's how to prepare emotionally for solo mining:

1. Accept that variance is the structure

Don't expect blocks at the mean. Expect them in clusters separated by gaps. The actual pattern of finding blocks looks chaotic; the underlying math is deterministic. Internalize that "long dry stretches" are normal, not failure.

2. Set time horizons that match the math

If your mean time is 133 days, don't make decisions at day 60. Don't change strategy at day 90. Don't quit at day 200. Plan for at least 2-3× mean time before evaluating performance. For a single S21+ on BCH, that's 9-12 months minimum.

3. Track outcomes against statistical predictions, not expectations

If after 200 days of mining you've found 0 blocks, you're in the bottom 22% of outcomes — unlucky but not extreme. If after 200 days you've found 3 blocks, you're in the top 5% — lucky. Either is statistically normal. Don't conflate "what I expected" with "what's normal."

4. Diversify across timescales and chains

Combine slow chains (BCH at 133-day mean per S21+) with fast chains (BC2 at 2-day mean per Bitaxe). The fast chains give you frequent reinforcement that the math works. The slow chains give you the bigger payouts. Don't put 100% on one chain — variance compounds in single-chain setups.

5. Consider scale

If your variance tolerance is low, scale up (more rigs reduces relative variance). If scale isn't available, mine on smaller chains where the math gives you blocks more often. Don't try to absorb high-variance experiences with insufficient scale and thin margins.

6. Hold through the dry stretches

The single most common mistake in solo mining is quitting during a dry stretch and missing the jackpot that arrives soon after. The math is brutal that way: the jackpot eventually arrives, but you might have already shut off your rig. Mine longer than your impatience suggests.

7. Don't seek causality where there is none

"I changed pools and immediately found a block" — that's variance, not causality. The pool you switched from was probably about to find one too. The math doesn't care about your strategic decisions; it cares about cumulative hashes contributed.

The math in everyday solo mining decisions

How do these probabilities affect actual decisions?

"Should I rent hashrate for a one-day burst?"

Compute the daily probability at the rented level. If 1 PH/s for 24 hours on BCH gives you ~3.2% probability of finding a block, decide whether $50-100 rental cost is worth that lottery ticket. Expected value: 3.2% × $1,400 = $44.80. If your rental cost is $50, EV is slightly negative; if $40, slightly positive.

"Should I switch from BCH to a smaller chain?"

Compute mean times on both. If your S21+ has 133-day mean on BCH ($1,400 reward) and ~2-day mean on BCH2 (~$10 reward), the expected daily revenue is roughly equal — but variance is wildly different. Choose based on your variance tolerance, not expected value, since EVs are similar.

"Should I keep mining after a 6-month dry stretch?"

Look at the cumulative distribution function. After 6 months on a 133-day-mean rig, you're at ~80th percentile of bad outcomes. The probability of finding a block in the next month is exactly the same as it was when you started. The dice have no memory. If your reasons to mine were sound when you started, they're still sound now.

"Should I add a second rig before the first finds a block?"

Yes, statistically. The marginal hashrate gives you more probability per unit time. The "wait until the first one hits" reasoning is gambler's fallacy — the first rig isn't "due."

Variance in fee revenue (a separate distribution)

Block subsidy is fixed per block — 3.125 BTC, 3.125 BCH, 1.81M XEC, etc. Transaction fees vary by block, sometimes wildly. This adds a second layer of variance:

• Most BCH blocks have fees of ~0.001-0.005 BCH (~$0.50-2.50)
• Some BCH blocks during high activity can have 0.1-0.5 BCH fees ($45-225)
• BTC blocks during Runes / Ordinals events have included 5-10 BTC of fees vs 3.125 BTC subsidy

If you find a block during a high-fee period, your reward could be 1.5-3× the base subsidy. If during a low-fee period, near-zero fees. Fee variance compounds with block-finding variance. Fortunately, fees are usually a small fraction (4-5%) of total reward, so the second-order variance is dominated by the first-order block-finding variance for almost all miners.

The ergodic argument

For mathematically inclined readers: solo mining is an ergodic process — the time-average of a single miner's outcomes converges to the ensemble-average across all miners, given sufficient time. In practice, this means: if you mine long enough (decades), your average revenue per year approaches the long-run expected value. The variance dominates short-term outcomes but vanishes in the long run.

The catch: "long enough" might be longer than human patience. For an individual miner with a 133-day-mean rig, the time horizon for variance to "wash out" to a few percent confidence interval is roughly 10-30 years. For a fleet of 100 rigs, it's 1-3 years. Scale dramatically shortens the convergence time.

Solo mining at small scale is fundamentally a long-time-horizon investment in the ergodic limit. If you can wait long enough, the math delivers. If you can't, you'll see the variance, not the expected value.

The coefficient of variation, by setup

One useful summary metric: the coefficient of variation (CV) measures how "spread out" yearly revenue is relative to the mean. Lower CV = more predictable. Higher CV = more lottery-like.

This is the practical guide to selecting your mining setup based on variance tolerance.

What the math doesn't tell you

The Poisson model is mathematically correct but assumes:

• Stable network hashrate — in reality, hashrate fluctuates 5-15% over any few-month period, slightly affecting your relative share
• Stable difficulty — difficulty adjusts roughly every 2 weeks; this changes your per-attempt probability slightly
• 100% uptime — every minute offline is missed lottery tickets
• No correlation between blocks — actually true for SHA-256, by design
• Stable BTC/BCH/XEC prices — for revenue projections; doesn't affect block-finding probability

None of these break the math significantly. They add noise, but the underlying Poisson structure remains the dominant story. The first-order math is right. Second-order corrections are real but small.

The kicker

Solo mining is not gambling. It is a high-variance honest probabilistic process governed by well-understood mathematics. The Poisson distribution describes block-finding. The exponential distribution describes time-between-blocks. The standard deviation scales with the square root of the mean. The expected value scales linearly. Everything that feels emotionally chaotic about solo mining is mathematically deterministic when viewed at the right scale.

The miners who internalize this math survive long dry stretches without panic, recognize lucky stretches without overconfidence, and eventually capture the expected value that the math promises. The miners who don't internalize it overreact to short-term outcomes, change strategies at exactly the wrong moments, and exit the system before the variance has time to wash out.

The math is on your side. The math is patient. The math doesn't reward your patience and doesn't punish your impatience — it just is what it is. The miners who learn to read it correctly are the ones who keep mining when others give up, and who collect the jackpots when they eventually arrive.

The owl knows that the field doesn't reward eagerness. Each night is a fresh roll. Some nights deliver. Most don't. The owl that hunts for ten thousand nights eats. The owl that hunts for ten gets discouraged. Pick your night count. The math handles the rest.