Average below threshold
np can lie below r while the upper tail remains operationally material; the mean is not a safe substitute for exceedance probability.
Probability
Binomial Event Risk Calculator
Calculate the exact probability of at least r adverse events across n independent exposures and translate that tail into a probability-weighted threshold consequence.
ADVERSE-EVENT TAIL SCREEN
Measure escalation probability at a count boundary, not only average events
The model builds the exact event-count distribution, sums all rows at or above the risk threshold, and keeps the exact threshold row and complementary probability visible.
P(adverse events >= threshold) -
Probability-weighted exposure -
Below-threshold probability -
P(exactly threshold) -
Expected adverse events -
Event-count variance -
LIVE DECISION RECORD
Threshold-neighborhood risk ledger
Rows around the escalation count show exact mass, cumulative position, classification, and consequence contribution.
| Adverse events | Exact probability | Cumulative probability | Risk classification | Weighted consequence row |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
X~Binomial(n,p); Risk=P(X>=r)=sum from k=r to nC(n,k)p^k(1-p)^(n-k); weighted exposure=Risk x L
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Independent exposure count - Fixed number of opportunities for the adverse event. | 75 |
| eventProbabilityPct | Adverse-event probability (%) - Stable per-exposure probability under the scenario. | 4.5 |
| riskThreshold | Risk threshold event count - Inclusive count r defining the escalation event X>=r. | 6 |
| thresholdExposure | Consequence at threshold - One conditional exposure amount for the threshold event. | 250000 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Operational-risk, quality, compliance, warranty, and portfolio teams screening a repeated-exposure count threshold.
Decision boundary: Use for a fixed count of independent equal-probability exposures and a binary event; severity by count and dependence require another model.
HOW TO DEFINE THE RISK EVENT
Five steps from exposure to escalation tail
- Define one exposure and the adverse event classification.
- Fix n for the decision period before observing outcomes.
- Estimate p from comparable exposures and document exclusions.
- Set an inclusive whole-number threshold tied to a real control or consequence.
- Review the tail complement and export the current threshold ledger.
TAIL-RISK FUNDAMENTALS
Five distinctions behind the risk number
- Exposure
- One opportunity for the defined adverse event.
- Threshold
- Smallest count classified as the risk event.
- Upper tail
- Combined mass for threshold and every larger count.
- Complement
- Probability below threshold; it must sum with risk to one.
- Consequence
- Conditional amount attached to the threshold event, separate from probability.
FORMULA AND DEFAULT SUBSTITUTION
Sum the escalation region exactly
P(X>=r)=sum from k=r to n of C(n,k)p^k(1-p)^(n-k)
At n=75 and p=0.045, expected adverse events are 3.375. With r=6, risk sums rows 6 through 75. Multiplying that probability by 250,000 gives the one-threshold weighted exposure while preserving exact event mass at k=6.
DEEPER RISK ANALYSIS
Three ways a tail can mislead
Consequence gradient
If seven events cost more than six, one flat threshold exposure is insufficient; each count needs its own severity in an outcome table.
Clustering
Shared causes or contagion violate independence and can move mass from central rows into both tails.
WORKED RISK CASES
Two thresholds with different meanings
Warranty escalation
Seventy-five installations each have a 4.5% claim chance. A six-claim review rule should be governed by P(X>=6), not by the expected 3.375 claims.
Zero-threshold audit
Setting r=0 returns 100% risk and the full consequence. This is an intentional boundary that confirms inclusive-tail logic.
RISK TERMINOLOGY
Six terms in the escalation record
- Adverse event
- The binary outcome counted as success by the mathematics.
- Expected count
- np, the center of repeated count outcomes.
- Risk threshold
- First whole count requiring escalation.
- Exact threshold mass
- Probability of X=r alone.
- Tail probability
- Probability of X>=r.
- Weighted exposure
- Tail probability times one conditional amount.
EVIDENCE RETENTION
Bind the threshold to its authority
Keep exposure roster, event definition, p source period, duplicate handling, threshold policy, consequence calculation, currency, and review owner. Record whether p was estimated before the current exposure period.
LIMITS AND EXCLUSIONS
Boundaries of the independent exposure screen
- n is fixed and exposures share one p.
- Event dependence and overdispersion are excluded.
- One consequence amount applies to the entire upper tail.
- Parameter uncertainty in p is not propagated.
- Regulatory and safety thresholds require authoritative domain methods.
RELIABLE SOURCES
Primary references for exact event tails
EVENT-RISK FAQ
Questions about thresholds and consequence
Why is the threshold inclusive?
The adverse event is defined as X>=r, so the exact r row belongs to risk.
Is expected event count the same as tail risk?
No. np is the distribution center; tail risk sums probabilities at and beyond a decision boundary.
Can the threshold be zero?
Yes. X>=0 is certain, useful as a boundary check but rarely a meaningful control.
Does weighted exposure equal expected loss?
Only when one entered consequence applies once to every threshold-reaching outcome.
What if exposures have different probabilities?
Use a Poisson-binomial or scenario-specific model; averaging p can distort a tail.
How does dependence affect risk?
Positive clustering often produces heavier tails than the independent binomial model.
IMPORTANT RISK NOTE
A small probability can still demand governance
Interpret tail probability alongside consequence, model uncertainty, and control effectiveness. This screen does not replace legal, safety, actuarial, or regulatory analysis.