To estimate your chance of finding at least one Bitcoin block while solo mining, calculate the expected blocks for your hashrate and time window, then convert that value with the Poisson probability formula. In JavaScript, use -Math.expm1(-expectedBlocks) rather than 1 - Math.exp(-expectedBlocks); it preserves precision when the probability is very small.
What the calculator estimates
The result is a probability for a specific time horizon, not a countdown or a promised block date. With sustained accepted hashrate H, fixed Bitcoin difficulty D, and interval t, expected work per block is approximately D × 232 hashes. The expected number of blocks in the interval is μ = Ht / (D × 232); the probability of finding at least one is 1 − e−μ. Bitcoin Wiki gives the multiplier 232 as 4,294,967,296 hashes per difficulty unit under this difficulty convention (Bitcoin Wiki Generation Calculator, last edited February 22, 2025).
The model assumes a constant rate and difficulty over the selected window. Actual delivered work, uptime, rejected or stale work, network conditions, and difficulty changes can alter real outcomes, especially over long intervals. A live difficulty estimate is a snapshot, not a prediction of future difficulty.
How to calculate the probability in JavaScript
Convert the miner’s rate to hashes per second and the duration to seconds. Keep difficulty positive, and use the same Bitcoin difficulty convention as the formula.
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function soloBlockProbability({ hashRateHps, intervalSeconds, difficulty }) {
for (const [name, value] of Object.entries({
hashRateHps,
intervalSeconds,
difficulty,
})) {
if (!Number.isFinite(value) || value < 0) {
throw new RangeError(`${name} must be finite and nonnegative`);
}
}
if (difficulty === 0) {
throw new RangeError("difficulty must be greater than zero");
}
if (hashRateHps === 0 || intervalSeconds === 0) {
return { expectedBlocks: 0, probabilityAtLeastOne: 0 };
}
const expectedBlocks =
(hashRateHps * intervalSeconds) / (difficulty * 2 ** 32);
const probabilityAtLeastOne = -Math.expm1(-expectedBlocks);
return { expectedBlocks, probabilityAtLeastOne };
}This returns both the expected block count and the chance of at least one block. They are different quantities: an expected count is an average over repeated equivalent intervals, not a percentage. Mining calculators use the Poisson model to express block-finding probability and expected time (SoloFury Solo Mining Calculator; D-Central Solo Mining Calculator).
Why use Math.expm1()?
For a small expected block count, Math.exp(-expectedBlocks) is very close to 1. Subtracting it from 1 can erase meaningful digits or round a nonzero probability to zero. MDN explains, “For very small values of x, adding 1 can reduce or eliminate precision.” The Math.expm1() method computes ex − 1 with improved precision for small x, making -Math.expm1(-μ) the stable equivalent of 1 - Math.exp(-μ) (MDN: Math.expm1(), last modified July 10, 2025).
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Validation and very small values
- Reject non-finite or negative hashrate, duration, and difficulty values; difficulty must be strictly positive for this formula.
- Zero hashrate or a zero-length interval means zero probability.
- For exceptionally large inputs, intermediate multiplication can overflow JavaScript’s finite-precision
Numberrange. For exceptionally small results, retain scientific notation or enough significant digits rather than formatting them as an unexplained0%. - Do not use
Number.EPSILONas a universal cutoff for tiny probabilities. It is 2−52, approximately 2.220446049250313 × 10−16, describing the spacing between 1 and the next larger representable JavaScript number (MDN: Number.EPSILON).
How to interpret the result and expected time
For a constant rate, the waiting time follows an exponential model. The mean waiting time is T = 1/λ, where λ = H / (D × 232) blocks per second. The median is T × ln(2), about 0.693 times the mean. After one mean waiting time, the probability of at least one block is about 63.2%, not 100%. Passing the mean time does not make a block due; a short interval can still contain a block, and a long wait can continue.
To present a result responsibly, show the time horizon, hashrate and units, difficulty value and snapshot time, expected blocks, and probability of at least one block. If you use a current network or difficulty estimate, identify its source and retrieval timestamp. Do not imply that a snapshot predicts conditions throughout a future interval.
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What this calculator does not tell you
This is a block-finding probability estimate, not a profitability calculation. It does not account for electricity, equipment, fees, taxes, uptime costs, or future network difficulty. Pool mining does not change the underlying probability that a hash meets the network target, but it changes how miners share payouts and the variance of individual payments. Bitcoin’s Developer Guide explains that pools set easier share targets and distribute proceeds through their payout systems, while solo miners attempt to find a network-valid block independently (Bitcoin Developer Guide: Mining).
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