Tail rows can drive consequence
The zero- and one-failure rows often hold most probability, but low-probability multi-failure rows can dominate contingency planning when each additional failure adds downtime.
Reliability
Component Failure Outcome Table Calculator
Build an exact binomial failure-count table with current consequence and expected-loss rows.
EXACT FAILURE OUTCOMES
Price every possible failure count in one observation window
For operations and reliability teams that need the complete binomial partition, current consequence values, and a visible dependence boundary rather than one average count.
Any failure-
Expected failures-
Expected loss/window-
Expected loss/all windows-
Consequence/failure-
CURRENT DECISION RECORD
Exact component-failure outcome table
Every row is regenerated from the active inputs and carried into Copy, TXT, and the page-specific PDF payload.
| Failures k | Probability (%) | Expected windows | Consequence |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
P(X=k)=C(n,k)p^k(1-p)^(n-k)
Waiting for valid inputs.
HOW TO USE
Five steps to an auditable outcome table
- Define identical components in one observation window.
- Enter per-component failure probability.
- Set the number of comparable windows.
- Enter repair and downtime consequence.
- Inspect every k row and reconcile probability and np.
BINOMIAL FUNDAMENTALS
Five ideas that make the rows complete
- Binomial state
- Exactly k failures among n components when every component has the same failure probability in one defined observation window.
- Outcome partition
- Rows from k=0 through k=n are mutually exclusive and exhaustive, so their probabilities must sum to one.
- Expected failure count
- The identity n x p is the long-run average number of failed components per comparable window, not the most likely row.
- Any-failure probability
- One minus the k=0 probability answers whether at least one component fails, regardless of how many fail.
- Consequence mapping
- Each row multiplies k by the entered per-failure repair and downtime consequence; shared losses need a separate model.
DEEP OUTCOME ANALYSIS
Read tails, expectations, and dependence separately
Expected windows are not scheduled events
A row value of 2.4 expected windows means the row averages 2.4 occurrences across many equivalent programs. It does not predict two events plus a partial event.
Dependence changes the distribution
Shared heat, vibration, power, software, or maintenance can make failures cluster. Positive dependence generally moves probability from middle rows toward the tails.
WORKED DECISION CASES
A normal operating case and a zero-risk boundary
Ten pump modules over 1,000 shifts
With n=10 and p=3%, the expected failure count is 0.3 per shift. The complete table shows how often no module fails, how often one fails, and the smaller multi-module tail used for spares and response planning.
Zero-probability validation boundary
When p=0, all probability belongs to k=0, expected failures and expected loss are zero, and every k>0 row must be zero. This is a direct reconciliation test for the table.
EVIDENCE RECORD
Preserve the definition behind p and each consequence
Retain the component population, observation-window definition, failure criterion, data period, exposure count, treatment of removals, repair-cost basis, downtime valuation, and independence review. A probability copied without these records cannot be reproduced or safely transferred to another population.
MODEL LIMITS
Conditions required by the exact binomial table
- Components are identical and independent within the observation window.
- Each component has one binary failure state per window; repeated repairs or recurrent failures are excluded.
- Consequences add linearly with k and do not include shared outages, repair overlap, or capacity thresholds.
- The same entered probability applies to every component and every reported window.
OUTCOME GLOSSARY
Six terms used in the result ledger
- Observation window
- The fixed mission, shift, cycle, or interval in which each component receives one binary failure opportunity.
- Bernoulli trial
- A component-level trial with two outcomes: failure or no failure during the window.
- Probability mass
- The probability assigned to one exact count k rather than to a continuous range.
- Combination count
- C(n,k), the number of distinct ways k failed components can be selected from n components.
- Expected loss
- The sum of each row consequence multiplied by that row probability.
- Tail outcome
- A low-frequency row with a failure count far from the distribution center.
FREQUENTLY ASKED QUESTIONS
Questions specific to component-count outcomes
Why does the table include zero failures?
The zero row is a real operating outcome and anchors the at-least-one calculation. Omitting it would prevent the probabilities from forming a complete partition.
Can components have different failure probabilities?
Not in this identical-binomial model. Use a Poisson-binomial or simulation model when component probabilities differ materially.
Does the consequence table include overlapping repairs?
No. It adds the entered per-failure consequence linearly. Shared mobilization, parallel repair, or capacity nonlinearities need explicit scenario logic.
Can observation windows overlap?
Overlapping windows can share the same failure exposure and violate independence across windows. Define non-overlapping comparable windows or document the dependence.
Why can expected windows be fractional?
Expected count is an average across repeated programs. Fractional values are meaningful for budgeting and capacity but are not literal partial incidents.
What if one component failure causes another?
The binomial independence assumption no longer holds. Use a dependency, common-cause, fault-tree, or event simulation model instead.
RELIABLE SOURCES
Probability and reliability references
IMPORTANT DEPENDENCE NOTE
An exact table can still answer the wrong model
The arithmetic is exact only for the entered identical independent probability model. Do not use the tail for common-cause, cascading, heterogeneous, or repeated failures without replacing that assumption.