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Strategic Conditioning

Most hard probability problems become easy once you condition on the right thing. Choosing what to condition on is the skill.

The law of total probability
P(A)=∑iP(A∣Bi)P(Bi)P(A) = \sum_i P(A \mid B_i)P(B_i)

Split into cases that cover everything and overlap nowhere, solve each, then weight by how likely the case was.

The law of total probability is the tool. The art is in picking the partition.

What makes a good choice

It creates independence. Variables that are entangled often become independent once you condition on a shared cause. Two stocks are correlated; conditional on the market return they may be nearly independent, which makes the joint problem tractable.

It reduces to a known case. Conditioning on the first step often leaves a smaller version of the same problem, which gives a recursion.

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