How should I practice Valid Range of a Three-Outcome Probability Parameter?

Start by reading the full statement, solve the visible examples, then compare the pattern against related IMC Trading phone screen problems in FastPrep.

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Valid Range of a Three-Outcome Probability Parameter

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Problem statement

Three mutually exclusive outcomes have probabilities q, weight*q^2 and 1-q-weight*q^2. Find all real values of q that make these three numbers a valid probability distribution.

For this exercise, assume the nonnegative integer weight is an input, generalizing the reported coefficient one. The valid set is one closed interval [0, upper]. Return its upper endpoint.

Complete maximumValidQ(weight). Answers within 1e-9 absolute error are accepted. No portfolio payoffs or trading decisions form part of this probability-range task.

Function

maximumValidQ(weight: int) → double

Examples

Example 1

weight = 1return = 0.6180339887498949

Nonnegativity requires q >= 0 and q+q^2 <= 1. Thus the reported valid interval is [0,(sqrt(5)-1)/2].

Example 2

weight = 0return = 1.0

The probabilities are q,0,1-q, so the valid range is [0,1].

Example 3

weight = 2return = 0.5

At q=0.5, the probabilities are 0.5,0.5,0. Larger values make the final probability negative.

Example 4

weight = 6return = 0.3333333333333333

The upper endpoint is 1/3 because 1/3+6*(1/3)^2=1.

Constraints

For this exercise, assume 0 <= weight <= 1000000.
Zero probabilities are allowed, so both interval endpoints are included.
Return the upper endpoint as a double.