Probability & Expected Value
valuation_probabilityCompute expected value and probability-weighted outcomes for startup scenarios: discrete E[X], joint probability of sequential events, probability-weighted value, VC portfolio expected return, Poisson event probability, and continuous E[X] over a range. Method selects the formula. Use for probability-weighted central estimates; for named bull/base/bear tables or option pricing use valuation_advanced, and to discount cash flows use valuation_time_value. Parameters apply per method: expected_value_discrete and probability_weighted need outcomes + probabilities; portfolio_return needs weights + returns; poisson needs mean_events + k; expected_value_continuous needs lower + upper. outcomes and probabilities must be equal length, and the probabilities should sum to 1. Routing: use valuation_advanced method 'scenario_analysis' for named bull/base/bear scenario tables, and its black_scholes/binomial methods for option pricing; use this tool for arbitrary outcome lists and probability-weighted central estimates. Only method is required; all other parameters are method-dependent, so supply those the selected method names and omit the rest (defaults apply where defined). Rate and decimal inputs are fractions (0.10 = 10%); probability and weight lists are in [0,1] and sum to 1. Returns value, method, inputs, assumptions, chapter, formula_number and calculation steps; pure arithmetic — no I/O and no external calls — rounded to 2 decimals, with no auth or rate limits. An unknown method, or a missing method-required parameter, returns an error instead of a value.
Input Schema
| Name | Required | Description | Default |
|---|---|---|---|
| k | No | Number of events k for the Poisson probability P(X=k); integer ≥ 0. | |
| lower | No | Lower integration bound (standard-normal domain, e.g. -1.0). | |
| upper | No | Upper integration bound (standard-normal domain, e.g. 1.0). | |
| method | Yes | Formula to apply. Options: expected_value_discrete = E[X] = Σ xᵢ·P(X=xᵢ) over a discrete outcome list.; joint_probability = P(total) = Π pᵢ for independent sequential events.; probability_weighted = E[V] = Σ pᵢ·Vᵢ.; portfolio_return = E[R] = Σ wᵢ·Rᵢ across a VC portfolio.; poisson = P(X=k) = e^-λ λ^k / k! for rare events.; expected_value_continuous = E[X] = ∫ x·f(x) dx over [lower, upper] on the standard normal. | |
| returns | No | Return of each asset or scenario as a decimal (0.20 = 20%), aligned with weights. | |
| weights | No | Portfolio or factor weights, each in [0,1] and summing to 1 (same order as the paired value list). | |
| outcomes | No | Possible outcome values x_i, in any currency unit (must match probabilities in length/order). | |
| mean_events | No | Poisson mean λ = expected number of events in the interval. | |
| probabilities | No | Probability of each outcome or stage, each in [0,1]; the list must sum to 1 where it is exhaustive. |
Output Schema
| Name | Required | Description | Default |
|---|---|---|---|
| error | No | Error message when the call fails. | |
| steps | No | Intermediate steps for traceability. | |
| value | Yes | Computed valuation or metric. | |
| inputs | No | Echo of the normalised inputs used. | |
| method | No | Formula / method name that produced the result. | |
| chapter | No | Source textbook chapter. | |
| assumptions | No | Modelling assumptions applied. | |
| formula_number | No | Source textbook formula number (e.g. '3.1'). | |
| defaults_applied | No | Optional parameters that were not supplied, so their documented defaults were used. |