= Law of total probability
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Let $(B_i)$ be a countable <measurable partition> of the <sample space>, with positive <probabilities>. For any <event> $A$, the law of total probability gives
$$
\mathbb P(A)=\sum_i\mathbb P(A\mid B_i)\mathbb P(B_i).
$$
The pieces $A\cap B_i$ are <disjoint events>, so add their masses and use the definition of <conditional probability>. Zero-mass partition pieces can be omitted without changing the answer.
For the coin selection in <probability>, let $B$ mean the $3/4$-head coin was selected and let $H$ mean the first toss was heads. Then $\mathbb P(H)=\mathbb P(H\mid B)\mathbb P(B)+\mathbb P(H\mid B^c)\mathbb P(B^c)=11/20$. Each conditional answer describes one possible hidden situation; its weight describes how often that situation occurs under the model. This decomposition is useful whenever a simple conditional model sits behind a more complicated marginal <probability distribution>.
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