Cutoff sensitivity
Moving r by one reclassifies an entire probability row, which can materially change both acceptance rate and expected value.
Probability
Binomial Event Outcome Table Calculator
Enumerate every binomial success-count outcome, apply an acceptance cutoff and separate accepted/rejected payoffs, and reconcile acceptance probability with expected decision value.
EXHAUSTIVE ACCEPT/REJECT STATES
Map every success count to a decision and payoff
The page applies one inclusive cutoff to the complete exact distribution, labels each row accepted or rejected, and sums probability-weighted payoff across all possible outcomes.
Acceptance probability -
Rejection probability -
Expected decision value -
Expected success count -
Most likely count -
Probability at modal count -
LIVE DECISION RECORD
Success-count decision outcome table
All count rows are exhaustive; the final column shows each row's contribution to expected decision value.
| Successes k | Exact probability | Cumulative probability | Decision | Row payoff | Expected-value contribution |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
P(K=k)=C(n,k)p^k(1-p)^(n-k); accept if k>=r; E[V]=sum over k of P(K=k)V(k)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Trial count - Fixed opportunities represented by the exhaustive outcome table. | 12 |
| successProbabilityPct | Success probability per trial (%) - Stable probability for each independent trial. | 62 |
| minimumAcceptedSuccesses | Minimum accepted successes - Inclusive decision cutoff: accept when k is at least this count. | 8 |
| acceptedPayoff | Payoff if accepted - Signed decision payoff applied to every accepted row. | 42000 |
| rejectedPayoff | Payoff if rejected - Signed decision payoff applied to every rejected row. | -9000 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Quality, procurement, experimentation, underwriting, and operations teams applying a count-based accept/reject rule.
Decision boundary: Use when every accepted row shares one payoff and every rejected row another; graded payoff, sequential rules, and dependence need richer decision analysis.
HOW TO BUILD THE DECISION TABLE
Five steps from outcome count to expected decision value
- Define a fixed set of independent binary trials.
- Enter the per-trial success probability from comparable evidence.
- Set the minimum accepted count before observing the batch.
- Assign signed payoffs to the accept and reject decisions on one basis.
- Inspect cutoff rows, probability closure, and expected-value contributions before exporting.
OUTCOME-TABLE FUNDAMENTALS
Five distinctions behind the rule
- Outcome state
- One possible total success count k.
- Decision rule
- Predefined mapping from count to accept or reject.
- Inclusive cutoff
- The exact threshold row is accepted.
- Decision payoff
- Consequence of acting on the classification, not value per success.
- Expected decision value
- Sum of every state probability times its assigned payoff.
FORMULA AND DEFAULT SUBSTITUTION
Classify first, then weight the payoff
V(k)=V_A when k>=r, otherwise V_R; E[V]=sum over k of P(K=k)V(k)
Defaults use n=12, p=0.62, and r=8. Rows k=8 through 12 receive 42,000; rows k=0 through 7 receive -9,000. The live model sums each row's probability-times-payoff contribution and reconciles accept plus reject to 100%.
DEEPER DECISION ANALYSIS
Three questions hidden by a single cutoff
Payoff realism
If barely passing and perfect performance have different consequences, two flat payoff states conceal value gradients.
Precommitment
Choosing the threshold after seeing the batch creates bias; the rule and payoff basis should be recorded in advance.
WORKED DECISION CASES
Two outcome tables with distinct governance meaning
Supplier pilot acceptance
Twelve independent lots each pass with probability 62%; the contract accepts at eight or more. The table shows the exact chance and expected payoff of applying that pre-agreed rule.
Cutoff above n boundary
Requiring 13 successes from 12 trials is rejected as an invalid rule rather than silently producing zero acceptance. That protects the audit record from an impossible specification.
DECISION TERMINOLOGY
Six terms in the outcome ledger
- Success count
- Total binary successes observed across n trials.
- Acceptance cutoff
- Minimum k mapped to the accepted decision.
- Acceptance probability
- Sum of exact masses in accepted rows.
- Rejection probability
- Complementary mass below the cutoff.
- Row payoff
- Signed consequence assigned to that decision state.
- Modal count
- Success count with greatest individual mass.
EVIDENCE RETENTION
Preserve the rule before outcomes arrive
Keep trial protocol, p source, cutoff approval, payoff rationale, currency, acceptance authority, exceptions, and decision date. Store the entire outcome table so the threshold row remains auditable.
LIMITS AND EXCLUSIONS
Boundaries of the two-payoff rule
- Trials are independent and share one p.
- The rule depends only on total success count.
- All accepted rows share one payoff and all rejected rows another.
- Sequential stopping, retesting, and appeal paths are excluded.
- Expected value does not replace legal, quality, or ethical acceptance criteria.
RELIABLE SOURCES
Primary references for exact outcome probabilities
DECISION-TABLE FAQ
Questions about cutoffs and row payoffs
How is this different from expected value per success?
Here payoff is attached to the final accept/reject decision, not accumulated for each success and failure.
Is the cutoff inclusive?
Yes. A row with exactly the minimum accepted successes receives the accepted payoff.
Why enumerate every row?
The table makes decision classification and each row's expected-value contribution auditable.
Can payoffs be negative?
Yes. Both accepted and rejected outcomes use signed values.
What if acceptance payoff varies by k?
This two-payoff model is insufficient; use a state-specific payoff table.
Does the most likely count determine the best decision?
No. Expected decision value uses all rows, and governance may depend on downside or constraints beyond the mode.
IMPORTANT DECISION NOTE
A probability table cannot choose the policy
The calculator evaluates an entered rule. Authority, fairness, quality standards, reversibility, and consequence definitions remain governance decisions outside the arithmetic.