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Probability, expectation and conditioning

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Start with the idea

An expectation weights each possible result by how often it occurs under the assumed model. Conditioning restricts the population before computing a frequency.

Symbols, units & horizon

X: random payoff
xⱼ: possible payoff j
pⱼ: probability of that payoff, nonnegative and summing to 1
E: expectation or probability-weighted average
P: probability
A, B: events
∩: intersection, both events
|: given, restrict to event B
Σⱼ: sum over outcomes

When and why to use this

Use expected payoff to compare strategies and conditional probability to assess what a signal adds.

Probability P(A) measures the chance of event A under a model. It lies between 0 and 1. Expectation E[X] is a probability-weighted average of a random quantity X; it is not a promise for the next trade. The vertical bar in P(A|B) means “given B.”

𝔼 [X] = \sum_{j} p_{j} x_{j}, P (A | B) = \frac{P (A \cap B)}{P (B)}

Algebra and arithmetic

Weight outcomes and restrict the denominator

For outcomes xⱼ with probabilities pⱼ, sum pⱼxⱼ. A +10 payoff with probability .6 and −5 with probability .4 has expectation 6−2=4.
If 12 of 20 signal days are wins, the conditional probability is 12/20=.6. Dividing counts by all observations in numerator and denominator gives the intersection formula.

Work it by hand

Expected payoff $4 does not mean the next trade earns $4; the example’s next payoff is either +$10 or −$5.

The intersection symbol ∩ means both events occur. A set lists possible outcomes; ∈ means “is a member of.” An indicator 1ₐ equals 1 when event A occurs and 0 otherwise. A hat, such as μ̂, marks an estimated rather than known parameter.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import isclose

def expectation(outcomes, probabilities):
    if len(outcomes) != len(probabilities) or not probabilities:
        raise ValueError("Matching nonempty inputs required")
    if min(probabilities) < 0 or not isclose(sum(probabilities), 1):
        raise ValueError("Probabilities must be nonnegative and sum to one")
    return sum(x*p for x, p in zip(outcomes, probabilities))

def conditional(joint_probability, conditioning_probability):
    if not 0 <= joint_probability <= conditioning_probability <= 1 or conditioning_probability == 0:
        raise ValueError("Require 0 <= joint <= conditioning <= 1, conditioning > 0")
    return joint_probability / conditioning_probability

print(expectation([10, -5], [.6, .4]), conditional(.12, .20))

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