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text: '$$\n\\newcommand{\\bP}{\\mathbb{P}}\n$$\n\n[summary: \n$$\n\\newcommand{\\bP}{\\mathbb{P}}\n$$\nWe say that two [event_probability events], $A$ and $B$, are *independent* when learning that $A$ has occurred does not change your probability that $B$ occurs. That is, $\\bP(B \\mid A) = \\bP(B)$. \nAnother way to state independence is that $\\bP(A,B) = \\bP(A) \\bP(B)$. ]\n\n\n\n\nWe say that two [event_probability events], $A$ and $B$, are *independent* when learning that $A$ has occurred does not change your probability that $B$ occurs. That is, $\\bP(B \\mid A) = \\bP(B)$. \nEquivalently, $A$ and $B$ are independent if $\\bP(A)$ doesn't change if you condition on $B$: $\\bP(A \\mid B) = \\bP(A)$. \n\nAnother way to state independence is that $\\bP(A,B) = \\bP(A) \\bP(B)$. \n\nAll these definitions are equivalent: \n\n$$\\bP(A,B) = \\bP(A)\\; \\bP(B \\mid A)$$\n\nby the [chain_rule_probability chain rule], so \n\n$$\\bP(A,B) = \\bP(A)\\; \\bP(B)\\;\\; \\Leftrightarrow \\;\\; \\bP(A)\\; \\bP(B \\mid A) = \\bP(A)\\; \\bP(B) \\ ,$$\n\nand similarly for $\\bP(B)\\; \\bP(A \\mid B)$.\n',
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