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1Repair Windows errors before they cause bigger problems2Fix the driver behind crashes, sound loss and screen glitches3Clear out junk files and repair common Windows errorsFor a miner hashing at a sustained rate H hashes per second, at Bitcoin difficulty D, over t seconds, the estimated chance of finding at least one block is -Math.expm1(-μ), where μ = H × t / (D × 232). Using Math.expm1() avoids losing precision when the probability is very small. This is a probability estimate under fixed-difficulty and sustained-hashrate assumptions—not a countdown or a promised block date.
What the calculator estimates
The calculation models block discoveries as a Poisson process. Under the standard Bitcoin difficulty convention, expected work per block is approximately D × 232 hashes; 232 is 4,294,967,296. For a constant effective hashrate, the estimated block-finding rate is λ = H / (D × 232) per second. Over an interval of t seconds, the expected number of blocks is μ = λt, and the probability of at least one is 1 − e−μ. The Bitcoin Wiki’s Generation Calculator describes the target-derived per-hash probability and expected-work relationship.
These are distinct outputs: μ is an expected block count, while 1 − e−μ is the chance of one or more blocks during the chosen interval. A value such as 0.2 expected blocks does not mean a 20% chance exactly; the at-least-one probability is 1 − e−0.2, approximately 18.1%.
JavaScript implementation
The function below accepts hashrate in hashes per second, interval in seconds, and a positive Bitcoin difficulty value. It returns a probability from 0 to 1. Validate user-facing units before calling it; for example, convert a duration in hours to seconds and a displayed rate in terahashes per second to hashes per second.
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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 0;
}
const expectedBlocks =
(hashRateHps * intervalSeconds) / (difficulty * 2 ** 32);
return -Math.expm1(-expectedBlocks);
}Keep both time and hashrate units consistent. If the user enters a hashrate in terahashes per second, multiply by 1012 before passing it to the function. Convert minutes, hours, or days to seconds before calculating. A unit mismatch can change the answer by many orders of magnitude.
Why use Math.expm1()?
The direct expression 1 - Math.exp(-expectedBlocks) subtracts two numbers close to 1 when the expected block count is tiny. Floating-point rounding can erase meaningful digits, or even display a nonzero chance as zero. JavaScript’s Math.expm1() computes ex − 1 with improved precision near zero, so the equivalent -Math.expm1(-expectedBlocks) is the stable form. MDN notes, “For very small values of x, adding 1 can reduce or eliminate precision.”
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Display small probabilities honestly
Keep the returned probability as a number for further calculations, and format it separately for readers. A percentage rounded to zero decimal places can hide a real but tiny nonzero chance. Show enough significant digits, or use scientific notation or a phrase such as “approximately 1 in N for this interval” when that presentation remains clear. Always label the interval and inputs alongside the result; do not display an unexplained “0%.”
How to read the result
Probability accumulates with elapsed time under the model, but it does not behave like a countdown. A miner can find a block quickly or go much longer than the average wait. For an exponential waiting-time model, the mean wait is T = 1/λ, while the median wait is T × ln(2), about 0.693 times the mean. After one mean waiting time, the probability of at least one block is 1 − e−1, about 63.2%—not 100%.
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Passing the expected waiting time does not make a block “due.” Each interval remains uncertain; the estimate says how likely a discovery is over that interval, not when one must happen. The equations for Poisson probability, expected blocks, mean time, and median time are also laid out by SoloFury.
Assumptions and limits
- Difficulty is fixed for the selected interval. The formula uses one difficulty snapshot. Difficulty can change, so longer-horizon results are conditional on holding it constant rather than forecasts of future network conditions.
- Hashrate is sustained effective work. The input should represent accepted hashing work, not merely a device’s advertised or nominal rate. Downtime, rejected shares, stale work, and variation in delivered hashrate can reduce realized work.
- The difficulty must belong to the relevant Bitcoin context. Do not reuse a Bitcoin difficulty or hashrate for another network or mining algorithm without an appropriate model.
- Live inputs need provenance. If an interface fetches a current difficulty or network estimate, identify the source and retrieval timestamp. A snapshot does not predict later difficulty.
D-Central’s calculator methodology likewise frames the estimate around a constant-difficulty method and its assumptions. A calculator should disclose its input snapshot and avoid implying that a stale value is current.
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What the calculator does not tell you
This result is not a profitability calculation. It does not include electricity, hardware cost, fees, taxes, uptime expense, or future difficulty. It estimates block-finding probability, not the value or net return of mining.
Pool mining also should not be compared to solo mining as though it changed the underlying probability that a hash meets the network target. A pool generally sets easier share targets and distributes proceeds through its payout system, changing payout frequency and variance. Solo mining leaves the miner attempting to find a network-valid block independently. See the Bitcoin Developer Guide’s mining explanation for shares and pool payout mechanics.
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Implementation checks before publishing a result
- Label hashrate units and normalize to hashes per second.
- Label the selected interval and normalize to seconds.
- Use a positive difficulty value and state which snapshot it represents.
- Show expected blocks separately from the chance of at least one block.
- Use
Math.expm1()for small-probability precision and avoid formatting a nonzero chance as an unexplained zero. - State that the result assumes sustained effective hashrate and fixed difficulty over the interval.
JavaScript uses finite-precision numbers. Number.EPSILON is 2−52, approximately 2.220446049250313 × 10−16, the spacing between 1 and the next larger representable number. It is not a universal tolerance for deciding whether arbitrary values are effectively zero; precision depends on the scale of the values being compared.
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