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Independence and Expectation

Independence

Two events are independent when knowing one occurred tells you nothing about the other:

Independence
P(B∣A)=P(B)⟺P(A∩B)=P(A) P(B)P(B \mid A) = P(B) \quad \Longleftrightarrow \quad P(A \cap B) = P(A) \, P(B)

Knowing A happened tells you nothing about B, which is exactly the case where the probabilities multiply.

Two flips of a fair coin are independent. Two cards drawn without replacement are not, because the first draw changes what remains.

Independence is an assumption you make about the world, and in markets it is usually the wrong one. Assets correlate, especially when it matters most: correlations that look comfortably low in calm conditions tend toward one in a crisis, which is precisely when a portfolio is relying on them to stay low. Assuming independence is what makes a risk model understate tail risk.

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