Mining Variance & Poisson Math

Solo mining variance is the gap between the long-run average time to find a block and what any single miner actually experiences. It is governed by the Poisson distribution: block-finding is random, independent, and memoryless, so dry stretches far longer than the so-called “mean time” are mathematically normal — not a sign of broken hardware, bad luck, or the wrong pool. This article shows the math behind that, with verified 2026 numbers.

Key takeaways

• ”Mean time to block” is not a countdown. For an exponential process, the median is only 69.3% of the mean — half of all miners find a block sooner, half later, with a long tail of unlucky outcomes.
• Roughly 1 in 8 single-rig miners find zero blocks in a year on a chain where their expected rate is ~3 blocks/year. That is the math working correctly, not a failure.
• Block-finding is memoryless. A 200-day dry stretch does not make you “due.” Your odds for the next 30 days are identical to your first 30 days.
• Scale cuts relative variance. Doubling hashrate doubles expected blocks but raises the spread by only ~41% — the coefficient of variation drops by ~29%.
• Solo mining is high-variance honest work, not gambling. There is no house edge: the protocol pays cryptographic work, your share is proportional to your hashrate.

A miner points a single high-end ASIC (~235 TH/s) at Bitcoin Cash. A calculator says the expected time to a block is around four months. They wait past four months and find nothing. Five months, still nothing. Six months — is the math broken? Is the rig faulty? Did they pick a bad pool? Almost always, the answer is none of the above. They are experiencing variance, and in solo mining that gap between expected value and short-term reality can be enormous.

Most solo mining advice handwaves this. “Mean time is X days” sounds like a deterministic timer. 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 you’re inside a perfectly normal slow tail of the distribution. Below: Poisson basics, the exponential time-to-block law, real percentile tables, a Monte Carlo simulation, and a framework for staying rational through the dry stretches.

Why is solo mining a Poisson process?

Mining hashes one number at a time, looking for a value below the network target. Each hash is statistically independent — SHA-256 makes the previous hash’s outcome irrelevant to the next attempt. That is the textbook setup for a Poisson process, which has three defining properties, all satisfied here:

1. Events occur independently. Finding (or not finding) a block doesn’t change the probability of the next attempt.
2. Events occur at a constant average rate. Over days, network difficulty and your hashrate are roughly stable, so the expected rate is constant.
3. Probability in any tiny interval is proportional to its length. Mine twice as long, get twice the chance of a block.

From these, the entire mathematics of solo mining variance follows:

• The number of blocks in time T follows a Poisson distribution with mean λT = (your_hashrate ÷ network_hashrate) × (T ÷ block_time).
• The time between consecutive blocks follows an exponential distribution with mean = 1/λ.
• The probability of zero blocks in time T is e^(-λT).
• The standard deviation of the block count in time T equals √(mean) — the square root of the expectation.

Memorize those four facts and the rest is application.

What does “mean time to block” actually mean?

”Mean time to find a block” is the most-cited and most-misunderstood number in solo mining. The mean is NOT what you should expect to wait. It is the long-run average across many trials, and single trials scatter widely around it. For the exponential distribution that governs time-between-blocks:

Median time = 0.693 × Mean time

Half of all single-trial outcomes finish in less than 69.3% of the mean; half take longer. The distribution is right-skewed — a long tail of unlucky outcomes drags the mean upward, which is exactly why the median sits well below the mean.

This percentile structure is universal and never goes stale, because it depends only on the multiple of the mean, not on any coin’s current difficulty:

Read the last row: even at four times the mean, there is still a 1.8% chance of zero blocks. The exponential has a fat right tail, so long dry stretches are mathematically guaranteed for some fraction of miners.

A worked example for June 2026. Bitcoin Cash network hashrate has been hovering near 3.4~4.0 EH/s (it swings between roughly 3 and 6 EH/s as SHA-256 miners rotate between chains — see the live figure on the CoinWarz BCH hashrate chart). At ~3.9 EH/s, a ~235 TH/s unit has a mean time to a BCH block of about 120 days. Applying the multiples above:

• ~50% chance of a block by day 83 (median)
• ~63% by day 120 (mean)
• ~78% by day 180
• ~90% by day 276
• ~95% by day 360
• ~5% chance you’ll still be waiting after a full year

Because that 120-day figure moves with network hashrate, treat the day-counts as a snapshot and pull the current numbers from the SoloFury solo mining calculator and the live difficulty view in the Network Radar. The percentile shape, though, is permanent.

Variance, standard deviation, and the one-year picture

For Poisson processes the variance equals the mean, so the standard deviation is √mean. Over a year of mining a single ~235 TH/s rig on BCH (~120-day mean):

• Expected blocks: 365 ÷ 120 = ~3.0 blocks
• Standard deviation: √3.0 = ~1.74 blocks
• ~68% of years land between roughly 1.3 and 4.8 blocks
• ~95% of years land between 0 and ~6.5 blocks

So in a typical year you might find 1, 2, 3, 4 or 5 blocks — all squarely normal. Finding 0 is unusual but not extreme; finding 7+ is unusual but not extreme. A single-year outcome from 0 to 7+ blocks is entirely consistent with the same underlying math.

A short hot streak proves the point in reverse. If a small fleet finds three blocks in 19 days, that annualizes to ~58 blocks/year — many standard deviations above its true expectation. That is not skill or a “hot” pool; it is a positive variance excursion, and somewhere in the same year there was almost certainly a 19-day window with zero blocks that nobody tweeted about. Clusters and gaps in roughly equal measure are the signature of the process.

Why do blocks cluster instead of spacing evenly?

Poisson processes have a counterintuitive property: events tend to cluster rather than arrive on a metronome. Finding a block today adds no “unluckiness” tomorrow — your probability for the next block is exactly what it was before. Over short windows this produces runs of blocks close together separated by long quiet stretches.

The pattern isn’t broken; it is 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, and doesn’t reward you for being patient. Each new attempt is a fresh roll. The clustering is what makes solo mining feel emotionally chaotic — months of nothing, then an abrupt jackpot, then more nothing — even though the underlying process is stationary.

Monte Carlo: 10,000 simulated mining years

Numbers are abstract, so simulate. Imagine 10,000 identical miners each running one ~235 TH/s rig on BCH for 365 days (a ~120-day mean, so ~3 expected blocks/year). The distribution of yearly outcomes — a Poisson distribution with λ ≈ 3.0 — looks like this:

Read it carefully. About 5% of single-rig miners find zero blocks in a given year — not because they did anything wrong, but because variance. About 30% find more than the mean. About 3.6% have a “jackpot year.” Total revenue across all 10,000 averages to the expected value, but individual experiences vary enormously. Some miners conclude “I have the magic touch”; others conclude “solo mining doesn’t work.” Both are reading too much into too few data points.

(Because the mean depends on network hashrate, the exact percentages shift as conditions move; the qualitative shape — a clump near 2~3 with a heavy upper tail — is stable.)

The Gambler’s Fallacy trap

Many solo miners fall into a classic error: “I haven’t found a block in 200 days, so I’m ‘due.’” This is false. Poisson processes are memoryless. The probability of a block in the next 30 days, given 200 dry days, equals the probability in any 30-day window:

P(block in next 30 days | 200 dry days) = P(block in any 30 days) = 1 − e^(−30/120) = 22.1%

The 200 dry days don’t help and don’t hurt — they simply don’t matter. Dice have no memory. The reverse holds too: finding a block last week doesn’t make you “less likely” this week. Lucky streaks aren’t punished; unlucky streaks aren’t compensated. Miners who internalize this hold steady; those who don’t change strategy during bad stretches and quit at exactly the wrong moment.

Does more hashrate reduce variance?

Yes — and this is where the math turns practical. Variance scales with the square root of the mean, but expected revenue scales linearly. Double your hashrate and:

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

Larger fleets experience proportionally less variance. The big farms aren’t lucky — they’re mathematically smoothed by scale. Illustrative figures at a ~120-day single-rig mean:

For a single rig, year-over-year block count can swing 55%+. For a 100-rig farm, swings are typically under 6%. Industrial mining is a lower-variance business than solo mining — not because the math changes, but because scale averages out the noise.

Is solo mining the same as gambling?

People equate the two because both involve probability and both have winners and losers. Mathematically they differ in the part that matters most — the structure:

Gambling (lottery, casino)

• House edge — the math is built to favor the operator over the long run.
• Negative expected value by design — players lose on average.
• Rigged probabilities — the structure guarantees the house wins.

Solo mining

• No house edge — the protocol issues block rewards for cryptographic work; nobody skims a structural cut (a transparent pool fee is operational, not structural).
• Positive or near-zero expected value depending on hardware and electricity — essentially “your hardware capability minus your costs.”
• Fair probabilities — your share of blocks is proportional to your share of network hashrate.

The variance is real; the structural unfairness isn’t. Solo mining is high-variance honest work. Gambling is low-variance dishonest work. The probability math rhymes; the structures are opposites. A 1% pool fee — versus the 2% many solo pools charge — is the only deduction, and it doesn’t change the fair, proportional nature of the odds.

Percentiles by hardware (June 2026 snapshot)

What the math looks like across the hardware spectrum. These depend on current network conditions; use the calculator for live figures.

Single ~235 TH/s ASIC on BCH (mean ~120 days)

Bottom line: ~95% chance of at least one block in a year, ~5% chance of none. The unlucky 5% aren’t doing anything wrong — they’re at the bad end of the distribution.

Single Bitaxe Gamma (~1.2 TH/s) on BTC (mean ~15,000 years)

With Bitcoin’s network near 980 EH/s, a single Bitaxe on BTC is pure lottery mode: the probability is non-zero but vanishingly small over any human timescale. Some Bitaxe operators have hit anyway, because thousands of units worldwide produce occasional jackpots even when each individual’s odds are tiny. For a realistic chance on the same hardware, the move is a smaller SHA-256 chain — see best coins to solo mine in 2026.

Smaller SHA-256 chains

On low-difficulty SHA-256 forks supported by SoloFury, a single modern ASIC — or even a Bitaxe — can have a mean time-to-block measured in days, not months. The same Poisson math applies; only the mean changes. That’s the value of multi-chain access: you pick the variance you can tolerate. Current per-chain means are in the Network Radar.

Variance in block reward (a second distribution)

The block subsidy is fixed: 3.125 BTC, 3.125 BCH, and on eCash a 3.125M XEC subsidy of which the block solver receives 58% (~1.81M XEC), with the rest going to the staking and developer outputs defined in the eCash protocol. Transaction fees, by contrast, vary block to block — adding a second layer of variance on top of block-finding.

• Most BCH blocks carry trivial fees (a fraction of a coin), because the chain runs far below capacity.
• On Bitcoin, fees are usually 1~5% of the reward — and recently under 1% in quiet stretches, according to on-chain data reported by The Block.
• But spikes are extreme. At the April 2024 halving, Bitcoin block 840,000 carried 37.6 BTC in fees on top of the 3.125 BTC subsidy — driven by the Runes launch, per Cointelegraph’s reporting on mempool.space data. Several blocks in that window exceeded 10 BTC in fees alone.

So a block found during a fee spike can be worth multiples of the base subsidy; one found during a quiet weekend is near-subsidy-only. Fee variance compounds with block-finding variance. For most miners most of the time, though, fees are a small slice of the reward, so first-order block-finding variance dominates the experience.

The ergodic argument

For the mathematically inclined: solo mining is an ergodic process — the time-average of a single miner’s outcomes converges to the ensemble-average across all miners, given enough time. Mine long enough and your revenue per year approaches the expected value. Variance dominates the short run and vanishes in the long run.

The catch: “long enough” can exceed human patience. For a single rig with a ~120-day mean, the horizon for variance to wash out to a few-percent band is on the order of 10~30 years. For a fleet of 100 rigs, it’s 1~3 years. Scale dramatically shortens convergence. Small-scale solo mining is fundamentally a long-time-horizon bet on the ergodic limit: wait long enough and the math delivers; otherwise you experience the variance, not the expected value.

What the math doesn’t capture

The Poisson model is exact under idealized assumptions, which reality bends slightly:

• Network hashrate isn’t perfectly stable — it can move 10%+ over a few months, shifting your relative share. BCH in particular sees sharp swings as miners rotate between SHA-256 chains.
• Difficulty adjusts — every block on BCH/XEC (ASERT) or roughly every two weeks on BTC, nudging your per-attempt odds.
• Uptime is rarely 100% — every minute offline is a missed ticket.
• Prices move — affecting revenue projections, though not block-finding probability.

None of these break the model; they add noise around a dominant Poisson structure. The first-order math is right. Second-order corrections are real but small.

How should you prepare emotionally for variance?

Once the math is on the table, the practical discipline follows:

1. Accept that variance is the structure

Don’t expect blocks at the mean. Expect clusters separated by gaps. The pattern looks chaotic; the math is deterministic. Internalize that long dry stretches are normal, not failure.

2. Set time horizons that match the math

If your mean is ~120 days, don’t make decisions at day 60, don’t change strategy at day 90, don’t quit at day 200. Plan for 2~3× the mean before judging performance — roughly 8~12 months for a single rig on BCH.

3. Track outcomes against statistical predictions, not hopes

Past the mean with zero blocks puts you in the unlucky-but-normal tail, not the broken zone. Don’t conflate “what I expected” with “what’s normal.”

4. Diversify across timescales and chains

Pair slow chains (big payouts, long means) with fast small chains (frequent reinforcement that the math works). Don’t put 100% on one chain — variance compounds in single-chain setups. The setup wizard can point one rig at multiple chains in minutes.

5. Match scale to your variance tolerance

Low tolerance? Scale up, or mine smaller chains where blocks arrive more often. Don’t try to absorb high-variance experiences with thin margins and a single machine.

6. Hold through the dry stretches

The most common solo-mining mistake is quitting during a dry stretch and missing the jackpot that arrives soon after. Mine longer than your impatience suggests.

7. Don’t invent causality

”I switched pools and immediately found a block” is variance, not cause. The math doesn’t care about your strategic decisions; it cares about cumulative hashes contributed.