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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $I$ be an index set, and let $A_i \\in \\mathcal{F}$ be an [[event|00SQGJNRTB]] for each $i \\in I$.\nThe events $(A_i)_{i \\in I}$ are mutually independent if for each finite nonempty subset $J \\subseteq I$,\n$$P\\left(\\bigcap_{j \\in J} A_j\\right) = \\prod_{j \\in J} P(A_j).$$\n\n## Remarks\n\nFor two events this is the [[independence of two events|0P0MZ4X682]].\nFor three events $A, B, C$ the definition asks for four equations: one for each of the pairs $\\{A, B\\}$, $\\{A, C\\}$, $\\{B, C\\}$ and one for the triple $\\{A, B, C\\}$.\nMutual independence implies [[pairwise independence|0N57D1XY6T]]. The converse is false: see [[pairwise independent events that are not mutually independent|0AR4B3K5QD]].\nIf we replace some of the events $A_i$ by their complements, the family stays mutually independent.\n\n## Explanations\n\n### L1 human\n\nFor many events, independence must hold for every group of them, not only for every pair.\nFor three tosses of a fair coin, the events \"toss $k$ gives heads\" for $k = 1, 2, 3$ are mutually independent, and all three happen with probability $1/8$.\n",
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