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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space.\nLet $B_1, B_2, \\dots$ be a finite or countable family of pairwise disjoint events with $\\bigcup_i B_i = \\Omega$ and $P(B_i) \u003e 0$ for each $i$.\nThen for each event $A$,\n$$P(A) = \\sum_{i} P(A \\mid B_i) \\, P(B_i).$$\n\n## Remarks\n\nThe events $B_i$ form a [[partition|09CCE86A6D]] of $\\Omega$ into events of positive probability. The proof uses $A = \\bigcup_i (A \\cap B_i)$, countable additivity and the [[multiplication rule|0BS3RMX8HM]].\nThe case of two events $B$ and $B^c$ is the [[law of total probability for an event and its complement|0W62TKMW7V]].\nExample: urn $1$ has $2$ red balls of $5$ and urn $2$ has $3$ red balls of $4$. Choose an urn with a fair coin and draw one ball. The ball is red with probability $\\frac12 \\cdot \\frac25 + \\frac12 \\cdot \\frac34 = \\frac{23}{40}$.\n\n## Explanations\n\n### L1 human\n\nSplit the world into cases that do not overlap.\nThe chance of $A$ is the average of its chances in the cases, weighted by the chances of the cases.\n",
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