Value asymmetry
A large payoff gap amplifies both expected upside and random batch-to-batch spread; probability and economics cannot be reviewed separately.
Probability
Binomial Event Expected Value Calculator
Combine a finite binomial event count with success value, failure value, and one fixed cost to obtain expected net value, outcome spread, and break-even success probability.
FINITE-TRIAL VALUE MODEL
Price both outcomes before averaging the event count
The model keeps success payoff, failure payoff, and batch cost explicit, then uses linearity of expectation to value the entire opportunity set and solve the probability needed to break even.
Expected net value -
Expected value per trial -
Break-even success probability -
Outcome-value standard deviation -
Expected successes -
Expected failures -
LIVE DECISION RECORD
Outcome-value reconciliation
Success, failure, and fixed-cost layers remain separate before the expected net is reconciled.
| Value layer | Expected count | Value per occurrence | Expected contribution | Calculation basis |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
E[V]=npv_s+n(1-p)v_f-F; sigma_V=|v_s-v_f|sqrt(np(1-p)); p_BE=(F/n-v_f)/(v_s-v_f)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Trial count - Number of repeated opportunities in the decision batch. | 500 |
| successProbabilityPct | Success probability per trial (%) - Stable probability of the payoff labeled success. | 18 |
| successValue | Value per success - Signed contribution from one success outcome. | 95 |
| failureValue | Value per failure - Signed contribution or loss from one failure outcome. | -14 |
| fixedCost | Fixed batch cost - One-time cost deducted after outcome expectation. | 6000 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Product, campaign, portfolio, operations, and experiment teams valuing repeated binary opportunities.
Decision boundary: Use for one stable event probability, two linear per-trial values, and one known batch cost; it does not model nonlinear capacity, learning, or correlated outcomes.
HOW TO BUILD THE VALUE CASE
Five steps from event semantics to break-even p
- Define the repeated trial and label the counted outcome unambiguously.
- Enter p from a comparable evidence base, not the desired business case.
- Assign signed value to both success and failure, including avoidable costs.
- Separate one-time batch commitment from per-trial economics.
- Review expected value, spread, and break-even probability together before exporting.
EXPECTED-VALUE FUNDAMENTALS
Five ideas that prevent payoff errors
- Linearity
- Expected batch value is n times expected per-trial value even though the realized count is random.
- Signed outcomes
- Both success and failure can carry positive or negative contributions.
- Fixed commitment
- A cost incurred once must not be multiplied by trial count.
- Break-even p
- The event probability at which expected net value equals zero.
- Outcome spread
- Count variability scaled by the value difference between success and failure.
FORMULA AND DEFAULT SUBSTITUTION
Show every value layer before netting
E[V]=npv_s+n(1-p)v_f-F
With n=500 and p=0.18, expected successes are 90 and expected failures 410. Outcome value is 90 x 95+410 x (-14)=2,810; deducting 6,000 gives -3,190. Break-even p solves 500[p x 95+(1-p) x (-14)]-6,000=0.
DEEPER VALUE ANALYSIS
Three questions beyond a positive average
Scale versus fixed cost
More trials spread a fixed commitment across opportunities, but only if the per-trial model remains stable at that scale.
Probability uncertainty
The calculation treats p as known. A decision should test plausible p values or retain an interval before calling the mean investable.
WORKED VALUE CASES
Two batches with different economic signals
Retention outreach batch
Five hundred contacts at an 18% save rate, +95 per save, -14 per non-save, and 6,000 setup cost produce a negative expected net. The break-even p identifies the evidence threshold for launching.
Equal-payoff boundary
If success and failure are both worth 20, changing p cannot affect value. Rejecting the break-even request prevents a meaningless division by zero.
VALUE TERMINOLOGY
Six terms in the economic record
- Expected successes
- n x p, an average count rather than a guaranteed integer.
- Failure value
- Signed contribution assigned to the complementary outcome.
- Expected per-trial value
- p v_s+(1-p)v_f before fixed cost.
- Fixed cost
- Known batch-level commitment deducted once.
- Break-even probability
- p making expected net equal zero.
- Value standard deviation
- Random count spread expressed in value units.
EVIDENCE RETENTION
Keep probability and payoff sources separable
Retain cohort definition, probability estimate period, success/failure accounting, variable-cost treatment, fixed-cost approval, currency, and decision horizon. Preserve sensitivity cases when p is uncertain.
LIMITS AND EXCLUSIONS
Boundaries of the two-outcome value model
- Trials are independent with one stable p.
- Values are linear and identical across trials.
- Fixed cost is known and occurs once.
- Capacity constraints, learning, discounting, and payoff distributions are excluded.
- Expected value alone does not establish affordability, liquidity, or risk tolerance.
RELIABLE SOURCES
Primary references for the event distribution
EVENT-VALUE FAQ
Questions about payoff, spread, and break-even
Does expected value predict the most likely profit?
No. It is a long-run probability-weighted average; a single batch can differ materially.
Can failure value be negative?
Yes. Enter signed values: revenue or benefit positive, loss or cost negative.
Why is fixed cost applied once?
The model treats it as a batch-level commitment. Per-trial costs belong inside both outcome values or an expanded model.
What if success and failure values are equal?
Then probability does not affect value and a break-even p is undefined, so the page rejects that case.
Can break-even probability fall outside 0-100%?
Yes. It means every valid p is profitable or no valid p can break even under the entered payoffs and fixed cost.
Does standard deviation include fixed-cost uncertainty?
No. Fixed cost is treated as known; spread comes only from the random success count.
IMPORTANT VALUE NOTE
A positive expectation is not a complete investment case
Review probability uncertainty, downside capacity, timing, dependencies, and governance constraints. The exported record documents assumptions; it does not authorize spending.