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Maintenance and Reliability Engineering
Failure Rate and Reliability Calculator
In short: the exponential model turns a constant failure rate into the probability a unit survives to a given time, R(t) = e^(−λt). Enter a failure rate, MTBF, or failures over time below and this tool returns R(t), the probability of failure, MTBF, the constant hazard, and the design life for a target reliability.
Calculate reliability from a failure rate
failure rate λ and mission time t → R(t) = e−λt, F(t), and design life for a target reliability
Reliability at the mission time
60.653% at t = 500 h
Conditional reliability is the same as R(t): the exponential model is memoryless, so a used unit is as good as new.
What this calculator computes
This tool works the exponential reliability model, the workhorse of constant-failure-rate analysis. From a single failure rate it answers the two questions engineers ask most: what is the probability a unit survives to a chosen time, and how long until reliability falls to a level you care about. The first is the reliability function R(t) = e^(−λt); the second is its inverse, the design life at which R(t) reaches a target. Around those it reports the probability of failure, the mean time between failures, and the constant hazard, and it plots the survival curve so the decay of reliability over the life of the item is visible at a glance.
The exponential model rests on one assumption, a constant failure rate, and that assumption is what makes it both simple and, used in the right place, powerful. When the hazard does not change with age, reliability follows a clean exponential decay governed by a single number, and that number, the failure rate, is exactly the reciprocal of MTBF. This is the flat middle of the classic bathtub curve, the random-failure regime that describes many electronic components and complex multi-part systems well. It is the natural companion to the MTBF calculator: the failure rate that tool reports is precisely the input this one turns into a survival probability.
What sets this calculator apart is that it goes both directions and surfaces the model’s defining quirk. Beyond the forward R(t), it solves the inverse design-life question that warranty and service-interval decisions actually need, and it states plainly that the exponential is memoryless, so conditional reliability equals R(t) regardless of prior age. You can enter the rate as a failure rate, an MTBF, or raw failures over operating time, and everything runs in your browser with nothing stored, alongside a chart and export.
How to use this calculator, step by step
Start by telling the calculator how you want to enter the failure rate. Choose the failure-rate option to type lambda directly in failures per hour, the MTBF option to enter the mean time between failures instead, or the failures-over-time option to let the tool derive the rate from a count of failures and the operating time they occurred in. All three lead to the same analysis; they simply match whatever data you have, whether a datasheet figure, a system rate, or your own maintenance records.
Then enter the mission time, the age at which you want the reliability, and optionally a target reliability between zero and one to get the design life. The calculator opens with a worked example already filled in, a failure rate of 0.001 per hour evaluated at 500 hours with a ninety-percent target, so you see a complete result immediately. Every field recomputes live, so changing the rate or the mission time updates R(t), F(t), and the curve at once, which makes it easy to explore how survival probability responds to a longer mission or a better component.
The result panel leads with the reliability at your mission time, then lists R(t), the probability of failure F(t), the failure rate, MTBF, the constant hazard, and the design life for your target. A note reminds you that conditional reliability equals R(t) because the model is memoryless. The chart plots the full R(t) survival curve so you can see the decay and locate your mission time on it, and you can download the analysis as a PDF or CSV or share it, all locally.
Reading R(t), F(t), and the hazard
The three functions describe the same failure behaviour from different angles, and reading them together builds intuition. The reliability function R(t) is the probability of survival to time t, a curve that begins at one and decays; the probability of failure F(t) is its mirror image, one minus R(t), the probability the unit has already failed by t, rising from zero. They always sum to one at any time, so a reliability of seventy percent at a given age is the same statement as a thirty-percent chance of failure by that age, and which you quote is just a matter of whether the survival or the failure framing is more natural for the decision.
The hazard rate is the third view and the most fundamental. It is the instantaneous rate of failure given survival so far, and for the exponential model it is constant and equal to the failure rate lambda at every age. This flatness is the whole content of the model: the chance of failing in the next instant never changes, which is why reliability decays at a steady proportional rate and why the curve is a clean exponential rather than the S-shape or steepening tail seen with aging. When you look at the constant hazard the calculator reports and recognise that it never moves, you are looking at the assumption that makes everything else follow.
It helps to connect the shapes to the numbers. At a mission time equal to the MTBF, reliability is always e to the minus one, about 37 percent, a useful landmark: the mean life is not the point of even odds but the point where only about a third survive, because the exponential is skewed. The median life, where half survive, is shorter than the mean, at about 0.693 times the MTBF. Noticing that the mean and the median differ is a good way to internalise that the exponential is not symmetric, and that quoting MTBF alone can mislead anyone who assumes it is a fifty-fifty age.
The memoryless property and conditional reliability
The single most distinctive feature of the exponential model is that it has no memory, and understanding this both sharpens your use of it and warns you where it fails. Because the hazard is constant, the probability that a unit lasts another t hours is the same no matter how long it has already run. A component that has operated flawlessly for a year faces exactly the same chance of surviving the next month as a brand-new one fresh from the box. Formally, the conditional reliability, the probability of surviving to time t plus s given survival to s, simplifies to plain R(t), independent of the elapsed age s.
This is why the calculator reports conditional reliability as identical to R(t) and notes that a used unit is as good as new. It is a genuinely useful simplification when it holds: for random failures that strike without regard to age, such as a voltage spike or a foreign-object event, prior survival really does tell you nothing about the future, and the memoryless model is exactly right. It also makes the mathematics of systems and spares tractable, because you never have to track the age of each part.
But the same property is the clearest signal of the model’s limits. Real components that wear out are emphatically not memoryless: an old bearing is far more likely to fail in the next hour than a new one, and pretending otherwise with an exponential model will badly understate the risk of an aging fleet. So when you find yourself doubting that a year-old unit is truly as good as new, that doubt is well founded and it is telling you to reach for the Weibull model, whose changing hazard can represent wear-out. The memoryless property is thus both the exponential’s convenience and its built-in warning label.
Five worked examples you can follow
Example 1: reliability at a mission time
The calculator opens with a failure rate of 0.001 per hour evaluated at 500 hours. Reliability is e to the minus 0.5, about 60.65 percent, so there is roughly a sixty percent chance the unit survives the 500-hour mission and a forty percent chance it fails during it. This is the basic forward calculation, and changing the mission time traces out the whole survival curve: at 1000 hours, equal to the MTBF, reliability drops to about 37 percent.
Example 2: the design life for a target
With a target reliability of ninety percent entered, the calculator reports the design life, the time by which reliability falls to that level, as about 105 hours. In other words, only after roughly 105 hours does the survival probability drop below ninety percent. This inverse calculation is what you use to set a warranty period or a service interval to a reliability goal, and it is the B10-style question, the age by which ten percent are expected to have failed.
Example 3: entering an MTBF instead
Switch the rate input to MTBF and enter 1000 hours; the failure rate becomes 0.001 per hour and every result matches the first example, because MTBF and failure rate are reciprocals. This mode is convenient when a datasheet quotes an MTBF directly, and it shows how the same reliability question can start from either the rate or its inverse without any change in the answer.
Example 4: deriving the rate from data
Choose the failures-over-time mode and enter four failures across 1116 operating hours, then evaluate at 360 hours. The failure rate is four divided by 1116, and reliability at 360 hours works out to about 27.3 percent, meaning under a seventy-three percent chance of surviving that long. This mirrors a common textbook case and shows how a handful of maintenance records becomes a reliability estimate, though with only four failures the estimate carries real uncertainty.
Example 5: the MTBF landmark
Set the mission time equal to the MTBF, whatever rate you have entered, and read the reliability: it is always about 37 percent. This landmark is worth remembering because it corrects a common misconception that the MTBF is the age by which half the units fail. In fact, by the mean life almost two-thirds have failed, because the exponential distribution is right-skewed, and seeing the same 37 percent appear regardless of the rate makes the point stick.
Three expert tips for reliable results
Check the constant-rate assumption first
The exponential only applies when the hazard is flat. If the item suffers infant mortality or wear-out, the answer will be wrong; use Weibull instead. Ask whether age changes the failure chance before trusting the model.
Keep units consistent
The failure rate and the mission time must share a time unit. A rate per hour with a mission in days, or a per-year rate with an hourly mission, silently produces a nonsense reliability.
Treat a data-derived rate as uncertain
A failure rate estimated from a few failures is a rough estimate, not a precise constant. The fewer failures behind it, the wider the true range, so read a low-count rate as indicative rather than exact.
The mathematics behind the results
The exponential model follows from a single premise: a constant hazard rate. If the instantaneous failure rate is a fixed lambda at every age, then the reliability function, the probability of surviving to time t, is the solution R(t) = e^(−λt), a pure exponential decay. The probability of failure is its complement, F(t) = 1 − e^(−λt), the cumulative distribution of the failure time, and the probability density of failure at time t is λ times e^(−λt). These three, the survival function, the cumulative failure function, and the density, are the complete description of the model, and all follow from the constant hazard.
The mean of this distribution, the mean time to failure or, for repairable items, the mean time between failures, is the integral of R(t) over all time, which evaluates to one over lambda. That is the exact reciprocal relationship the calculator uses to move between the rate and the MTBF.
The design life for a target reliability inverts the survival function: setting e^(−λt) equal to the target R and solving gives t = −ln(R)/λ, which is what the calculator computes when you supply a target.
And the memoryless property falls straight out of the algebra: the conditional reliability R(t+s given s) is R(t+s)/R(s), and for the exponential that ratio is e^(−λ(t+s))/e^(−λs) = e^(−λt) = R(t), free of the elapsed age s.
Two assumptions govern whether these clean results apply. The first is the constant hazard itself, exact only for the exponential and a good approximation only in the random-failure regime; when the true hazard rises or falls with age, the correct model is the Weibull, of which the exponential is the special case with shape parameter equal to one. The second is that failures are independent, so that combining rates by addition, as system reliability does, is valid. The calculator computes the exponential results exactly; judging whether the exponential is the right model for your item is the analyst’s job, and the memoryless note is there partly as a prompt to make that judgement.
Where the exponential model is used
The exponential reliability model is a fixture of reliability engineering wherever failures are, to good approximation, random rather than age-driven. It is the standard model for many electronic components, whose failures during useful life are dominated by random events rather than wear, and it underlies the failure-rate handbooks and prediction standards that tabulate component rates for building up a system figure. Because independent failure rates add, the exponential is also the natural building block for system reliability: a series system of exponential components is itself exponential with a rate equal to the sum, which keeps large reliability models tractable.
It also connects tightly to the rest of this silo. The failure rate here is the reciprocal of the MTBF from the MTBF, MTTR and availability calculator, so the two tools are two views of the same reliability.
The rate is the input the system reliability calculator combines across components, and the design life this tool computes for a target reliability feeds directly into warranty terms and preventive-maintenance intervals.
Where the exponential stops being appropriate, because the hazard is changing with age, the Weibull analysis calculator takes over, and recognising that handoff is one of the main judgements the constant-hazard model forces you to make. Return to the Maintenance and Reliability hub for the full set.
Beyond maintenance, the same model shows up in service-level and safety work. In IT and telecommunications the exponential underlies the failure-rate side of the availability targets written into service-level agreements, where component rates roll up into a system rate and then into expected downtime.
In functional safety, the constant-hazard assumption is built into the way random hardware failure rates are combined and compared against the failure-rate budgets that safety-integrity levels impose. In warranty engineering, the design-life calculation this tool performs, the time at which reliability falls to a chosen level, is exactly how a warranty period is aligned with an acceptable field-return rate.
Across all of these the appeal is the same: one number, the failure rate, captures the behaviour, and the exponential turns it into a survival probability, a downtime expectation, or a warranty term as the situation requires.
Estimating a failure rate you can trust
A reliability answer is only as good as the failure rate behind it, and there are three common sources, each with its own caveats. The cleanest is your own operating data: count the failures over a known operating time and divide, exactly as the failures-over-time mode does. The catch is statistical, because a rate from a small number of failures is uncertain; four failures give a very different confidence than four hundred, and a single failure barely constrains the rate at all. Treat a low-count estimate as a rough central value, not a precise constant, and gather more operating time before betting heavily on it.
The second source is a manufacturer datasheet or a reliability-prediction handbook, which quotes a failure rate or MTBF for a component under stated conditions. These are useful starting points, but they are conditional on the environment, temperature, and duty assumed in the source, and real conditions can shift the rate substantially, so a datasheet MTBF should be treated as an optimistic baseline unless your conditions match the test. The third source is a combined system rate built by adding component rates, which is valid for independent failures in series but assumes every contributing rate is itself sound.
Whatever the source, the discipline is to be honest about the constant-rate assumption. The failure rate is a single number that summarises a whole life only if the hazard really is flat over the range you care about. If the component is early in life and still shedding infant-mortality defects, or late in life and wearing out, a single rate averages over a changing reality and the exponential prediction will be off. The practical check is to ask whether a used unit really is as good as new; if the honest answer is no, the rate is not constant and the Weibull model is the better home for your data.
When to switch to the Weibull model
The exponential model is a special case of the more general Weibull model, and knowing the boundary between them is central to using either well. The Weibull adds a shape parameter that lets the hazard rise, fall, or stay flat; when the shape equals one, the hazard is constant and the Weibull is exactly the exponential. So the question of whether to use the exponential is really the question of whether the shape is one, that is, whether the failure rate is genuinely age-independent. For the flat middle of the bathtub curve it effectively is, and the exponential’s simplicity is a real advantage.
Two regimes break the assumption. Early in life, infant mortality dominates: manufacturing or installation defects cause a high initial failure rate that falls as weak units are weeded out, a decreasing hazard the exponential cannot represent and that calls for a Weibull shape below one. Late in life, wear-out dominates: fatigue, corrosion, and erosion drive an increasing hazard, a Weibull shape above one, and here the exponential is dangerously optimistic because it ignores aging entirely. Using a constant rate on a wearing-out population will predict far more survivors than reality delivers, which is exactly the kind of error that leads to missed maintenance and in-service failures.
The practical rule is to use the exponential when failures are random and age-independent, and to move to Weibull the moment the data or the physics say the hazard is changing. The clearest tell is the memoryless property: if treating a used unit as good as new feels wrong for your item, the hazard is not constant and the exponential is the wrong model. This calculator computes the exponential exactly and flags its memoryless nature precisely so that you can recognise when that nature no longer fits, and make the switch to Weibull deliberately rather than by accident.
Reliability and availability are different questions
It is easy to conflate reliability with availability, but they answer different questions and the exponential model speaks only to the first. Reliability, R(t), is the probability of surviving without any failure up to time t; it is about not failing at all over a window. Availability is the fraction of time an asset is up over the long run, and it allows for failures as long as repairs are quick, so a machine that fails often but is restored in minutes can have high availability and low mission reliability at the same time. The two coincide only for items that are never repaired, where a single failure ends the story.
The distinction matters when you choose which number to quote. For a mission that must run uninterrupted, a satellite pass, a batch that cannot be paused, a flight, reliability over the mission window is the right measure, because a single failure is a failure regardless of how fast it could be fixed.
For a production line judged on long-run output, availability is what counts, because brief stoppages are absorbed. This calculator computes reliability from the failure rate; the companion MTBF and availability calculator takes the same failure behaviour and, adding repair time, produces availability.
Using the two together, one for mission reliability and one for long-run uptime, keeps you from answering the wrong question with the right arithmetic.
The exponential and the bathtub curve
The constant failure rate that the exponential model assumes is not a universal truth about equipment; it is a description of one phase of life. The classic bathtub curve plots the hazard rate over the whole life of a population and shows three regions: a falling rate early on as infant-mortality defects are weeded out, a long flat middle where failures are random, and a rising rate at the end as wear-out sets in. The exponential model describes only that flat middle, the useful-life period, and it is exactly there that treating the failure rate as constant is justified and the memoryless property genuinely holds.
Seeing where the model sits on the curve clarifies both its power and its limits. During useful life, failures arrive from random external causes rather than from aging, so a single rate captures the behaviour well and the clean exponential mathematics applies.
But the same picture warns that the flat middle is bounded on both sides, and that applying the exponential to a population still in infant mortality or already into wear-out imports a constant-rate assumption the physics does not support.
Good practice is to establish, from data or engineering knowledge, that the item is in its useful-life phase over the horizon you care about before trusting an exponential prediction, and to watch for the upturn of wear-out that signals the end of the flat region and the point where scheduled replacement or a Weibull analysis should take over.
A short history of the exponential model
The exponential distribution became the foundation of reliability engineering in the years after the Second World War, when the growing complexity of electronic and military systems made their failures a first-order concern.
Early reliability work on vacuum-tube equipment found that, once early defects were screened out, failures during service arrived at a roughly constant rate, and the exponential distribution, with its single parameter and memoryless property, fit that behaviour and was mathematically tractable at a time when computation was expensive.
It was adopted into the military reliability-prediction standards that tabulated component failure rates, and the reciprocal link between failure rate and MTBF entered the everyday vocabulary of engineers from there.
That history explains both the model’s dominance and the cautions that grew up around it. The exponential was so convenient, and so well suited to the random-failure electronics of the era, that it was sometimes applied indiscriminately, including to mechanical parts that plainly wear out, which prompted the reliability community to stress the bathtub curve and the importance of checking the constant-rate assumption.
The rise of Weibull analysis, which generalises the exponential to a changing hazard, was in large part a response to that overuse.
Today the exponential is understood as the right tool for one well-defined situation rather than a universal law, and this calculator reflects that by computing the exponential exactly while flagging, through the memoryless note and the pointers to Weibull, the boundary of where it applies.
Common mistakes to avoid
A few errors recur when people apply the exponential reliability model. Watch for them.
- Using it on wear-out or infant-mortality items. The constant rate is wrong outside useful life; a worn part is not memoryless. Switch to Weibull when the hazard changes with age.
- Reading MTBF as a fifty-fifty age. At the MTBF only about 37 percent survive, not half; the median life is shorter, about 0.693 times the MTBF.
- Mismatched time units. The failure rate and the mission time must use the same unit; a per-hour rate with a mission in days gives a meaningless reliability.
- Over-trusting a rate from few failures. A rate from one or two failures is highly uncertain; treat it as a rough central value, not a precise constant.
- Confusing reliability with availability. R(t) is the chance of no failure over a window; availability allows failures with quick repair. Quote the one your decision needs.
- Ignoring the environment behind a datasheet rate. A quoted MTBF assumes stated conditions; harsher temperature or duty can raise the real rate well above it.
Input format and quick reference
Pick how you enter the rate: a failure rate per hour, an MTBF, or a count of failures over operating time. Then enter the mission time and, optionally, a target reliability between 0 and 1 for the design life. Keep the rate and the mission time in the same time unit. The reference below explains each output.
| Output | What it means |
|---|---|
| Reliability R(t) | Probability of surviving to the mission time, e^(−λt) |
| Probability of failure F(t) | Probability of having failed by t, equal to 1 − R(t) |
| Failure rate λ | Failures per hour, equal to 1 / MTBF |
| MTBF | Mean time between failures, 1 / λ |
| Hazard rate | Instantaneous failure rate; constant and equal to λ for this model |
| Design life for target | Time at which reliability falls to the target, −ln(R)/λ |
Frequently asked questions
What is the failure rate?
The failure rate, usually written as the Greek letter lambda, is the number of failures expected per unit of operating time. It answers how often, on average, a component fails while it is running, and it is the most compact way to state reliability. Under the constant-hazard assumption that defines the exponential model, the failure rate does not change with age, and it is exactly the reciprocal of the mean time between failures, so a failure rate of 0.001 per hour is an MTBF of a thousand hours. You can estimate it from data by dividing the number of failures by the total operating time, and this calculator accepts it directly, as an MTBF, or as failures over time.
What is the reliability function R(t)?
The reliability function R(t) gives the probability that a unit survives, without failing, up to a chosen time t. For the exponential model it is R(t) = e^(−λt), a curve that starts at one when t is zero and decays smoothly toward zero as time grows, with the failure rate lambda setting how fast it falls. Reading it is straightforward: R(500) of 0.6 means a 60 percent chance the unit is still working after 500 hours. This calculator reports R(t) at the mission time you enter and plots the whole curve, so you can see both the specific answer and how survival probability declines over the life of the item.
What is the probability of failure F(t)?
The probability of failure F(t), also called the unreliability, is the complement of reliability: F(t) = 1 − R(t), the chance the unit has failed by time t. If reliability at 500 hours is 60 percent, then unreliability is 40 percent, the probability it has failed at least once by then. F(t) is the cumulative distribution function of the failure time, rising from zero toward one as time grows, and it is the natural quantity when you care about the chance of failure within a warranty or mission window rather than the chance of survival. The calculator reports both R(t) and F(t) so you can read whichever framing fits your question.
What does a constant failure rate mean?
A constant failure rate means a unit is no more or less likely to fail as it ages: its chance of failing in the next hour is the same whether it is new or has been running for a year. This is the defining property of the exponential model and describes the flat, random-failure middle of the classic bathtub curve, after early-life defects have been screened out and before wear-out sets in. It is a good approximation for many electronic components and for complex systems made of many parts, but it is wrong for items dominated by infant mortality or wear-out, which need the Weibull model instead.
What is the memoryless property?
The memoryless property is the surprising consequence of a constant failure rate: the probability that a unit survives an additional t hours does not depend on how long it has already run. A component that has worked for a thousand hours has exactly the same chance of lasting another hundred as a brand-new one, because the hazard never changes.
Formally, the conditional reliability R(t given survival to s) equals R(t), the same exponential regardless of the prior age s.
This is why the exponential model is often described as treating a used item as good as new, and it is both its greatest convenience and the clearest sign of when it does not apply, since real parts that wear out are certainly not memoryless.
What is design life or the B-life for a target reliability?
The design life for a target reliability is the time at which reliability falls to a chosen level, the inverse of the R(t) question. If you want to know how long a unit lasts until only ninety percent are expected to still be working, you solve R(t) = 0.9 for t, giving t = −ln(0.9)/λ.
This is closely related to the B-life notation used in bearings and other components, where B10 life is the time by which ten percent have failed, that is, the design life for ninety percent reliability.
The calculator computes this design life whenever you enter a target reliability, which turns the model around from predicting reliability at a fixed time to sizing a warranty or service interval to a reliability goal.
How is the failure rate related to MTBF?
For the exponential model the relationship is exact and simple: the failure rate is one divided by the mean time between failures, and MTBF is one divided by the failure rate. A failure rate of 0.002 per hour is an MTBF of 500 hours, and vice versa. This reciprocal link is why the two are used interchangeably in constant-hazard reliability work, and it is how a failure rate estimated from a datasheet or a combined system rate is turned into the more intuitive MTBF, or the other way around. The calculator lets you enter either one and reports both, along with the reliability function they imply.
When should I not use the exponential model?
Avoid the exponential model whenever the failure rate is not roughly constant, which is precisely when aging matters.
Equipment in its infant-mortality phase, where weak units fail early and the rate falls with time, and equipment in wear-out, where the rate climbs as parts fatigue or corrode, both violate the constant-hazard assumption, and using the exponential there will misstate reliability, often badly.
The Weibull model, with its shape parameter that captures a rising or falling hazard, is the right tool for those regimes. Use the exponential for the random-failure middle of life, for many electronic components, and for complex systems, and switch to Weibull when the data show the rate changing with age.
Does this calculator store the numbers I enter?
No. The calculator runs entirely in your browser. The failure rate, MTBF, times, and other values you enter are never sent to our servers, stored, or shared. You can download a PDF or CSV of your results locally, and nothing leaves your device. See our Privacy Policy for details.
Is the failure rate and reliability calculator free?
Yes. The failure rate and reliability calculator is completely free, with no account, sign-up, or usage limit. It returns the failure rate, MTBF, reliability R(t), probability of failure F(t), the constant hazard, and the design life for a target reliability, along with the R(t) curve and PDF and CSV export at no cost.
Related maintenance and reliability calculators
More tools in this silo. Return to the Maintenance and Reliability hub for the full set.
Sources, disclaimer and editorial transparency
This calculator applies the exponential (constant-hazard) reliability model: R(t)=e^(−λt), F(t)=1−R(t), constant hazard h(t)=λ, MTBF=1/λ, and design life t=−ln(R)/λ for a target reliability, consistent with standard reliability engineering references. This calculator and guide are created and reviewed by the OpsCalculators team; see our Editorial Policy for how each tool is researched, built, and tested.
Results are accurate estimates for planning and education, not certified reliability or safety engineering advice, and they assume a constant failure rate (a flat hazard) and independent failures. For equipment in infant-mortality or wear-out, where the hazard changes with age, use Weibull analysis. See our full Disclaimer. OpsCalculators.com is operated by MAFHH INTERNATIONAL LTD. Your data is processed in your browser and never stored; see our Privacy Policy.