Free lesson · Prediction strategies

Bounds for joint and union events

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Even without an independence assumption, marginal probabilities restrict how large or small an overlap can be.

The joint cannot exceed either marginal event. Its lower bound is determined by the fact that total union probability cannot exceed one. These are probability coherence bounds before execution frictions.

When market prices are substituted for probabilities, discounting, spreads and funding matter. A probabilistic inconsistency points to a candidate state portfolio only after exact contracts and tradability are checked.

Apply it in a strategy

• Detect incoherent forecast combinations and construct stress scenarios without inventing independence.
• Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
• Stress this failure condition: Coherent point probabilities do not determine dependence, and tradable contract prices need not be undiscounted probabilities.

Research deliverable

Build and explain a bounds for joint and union events worksheet. Detect incoherent forecast combinations and construct stress scenarios without inventing independence.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def joint_bounds(a,b):
    if not 0<=a<=1 or not 0<=b<=1: raise ValueError("Invalid marginal probability")
    return max(0,a+b-1),min(a,b)

print(joint_bounds(.70,.60))