Probability
Calculate expected payoff, standard deviation, loss probability, and discrete percentile outcomes for a two-branch, four-leaf decision tree.
TREE PAYOFF DISTRIBUTION
This calculator turns four terminal decision-tree leaves into a discrete payoff distribution. Decision analysts can compare expected payoff, spread, loss probability, and stepwise percentile outcomes, while seeing the exact probability and contribution of every leaf.
TREE PAYOFF DISTRIBUTION
Use the distribution ledger to determine whether an attractive expected payoff is supported broadly or depends on a small high-payoff branch that leaves material downside elsewhere.
| Terminal path | Probability mass (%) | Payoff | Expected-value contribution | Loss classification |
|---|
CURRENT CALCULATION PROCESS
pi = product of path probabilities; E[X] = sumpi xi; sigmaX = sqrtsumpi(xi - E[X])^2; qalpha = inf{x: F(x) >= alpha}
The model multiplies probabilities along each terminal path, pairs each path mass with its signed payoff, and sums the contributions for expected value. Spread is the probability-weighted squared distance from that mean. Percentiles are selected from the payoff-sorted cumulative mass, so they must equal an actual leaf payoff.
HOW TO USE THIS MODEL
TREE PAYOFF DISTRIBUTION FUNDAMENTALS
MODEL AND FORMULA
The model multiplies probabilities along each terminal path, pairs each path mass with its signed payoff, and sums the contributions for expected value. Spread is the probability-weighted squared distance from that mean. Percentiles are selected from the payoff-sorted cumulative mass, so they must equal an actual leaf payoff.
DEEPER ANALYSIS
Two trees can share the same expected payoff while one concentrates near the mean and the other alternates between a large gain and a severe loss. Standard deviation, loss mass, and terminal ledger reveal that difference.
When cumulative probability crosses 10% or 50% at a single leaf, a small probability revision can make the reported quantile jump to another payoff. Preserve unrounded path probabilities and inspect boundary sensitivity.
Cash, utility, net present value, and mission score are different constructs. Combining them without an explicit transformation creates a number with no defensible interpretation even when the arithmetic is exact.
WORKED DECISION CASES
The default tree has expected payoff 59.6 but 40% probability on negative leaves. A sponsor who only sees the mean may approve too much exposure; the full ledger prompts review of the -30 and -10 outcomes and whether controls can reduce their path probabilities.
An alternative option can show a favorable expected payoff because a low-probability leaf carries an enormous gain. If the median and 90th percentile remain modest, the organization may prefer a less skewed option despite the higher arithmetic mean.
TECHNICAL LANGUAGE
EVIDENCE AND DATA LINEAGE
Keep branch counts or elicitation records, conditional-outcome evidence, payoff worksheets, valuation date, discounting convention, currency, time horizon, and the rule for classifying negative outcomes. Verify that probabilities sum through the tree and that payoffs include comparable cost and benefit categories. Record expert dependencies when probabilities or consequences come from the same judgment source.
LIMITS AND EXCLUSIONS
RELIABLE SOURCES
FREQUENTLY ASKED QUESTIONS
The median depends on cumulative probability after sorting payoffs. Several lower-payoff leaves can combine so that the cumulative mass first reaches 50% at that leaf.
No. Expected value can be positive while loss probability or worst-case consequence remains unacceptable. Review downside measures and decision constraints separately.
The labels preserve the event-tree meaning. A "success" leaf can still have a poor payoff under some value definitions, so payoff classification is calculated independently from the outcome label.
Not directly. Convert to a common valuation basis, price date, horizon, and risk convention before comparison.
A discrete CDF advances by entire leaf masses. Crossing a percentile threshold transfers the quantile to another terminal payoff rather than moving it continuously.
Simulation becomes useful with many branches, dependent variables, continuous payoff distributions, or repeated stages. For four fixed leaves, exact enumeration is clearer and free of simulation error.
IMPORTANT NOTE
The calculations exactly summarize the entered four-leaf distribution. They do not prove that all material pathways are represented, that probabilities are calibrated, or that signed payoffs capture legal, safety, strategic, liquidity, or human consequences.