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text: 'Let's say that, in a certain forest, there are 2 sick trees for every 3 healthy trees. We can then say that the odds of a tree being sick (as opposed to healthy) are $(2 : 3).$\n\nOdds express *relative* chances. Saying "There's 2 sick trees for every 3 healthy trees" is the same as saying "There's 10 sick trees for every 15 healthy trees." If the original odds are $(x : y)$ we can multiply by a positive number $\\alpha$ and get a set of equivalent odds $(\\alpha x : \\alpha y).$ \n\nIf there's 2 sick trees for every 3 healthy trees, and every tree is either sick or healthy, then the *probability* of randomly picking a sick tree from among *all* trees is 2/(2+3):\n\n\n\nIf the set of possibilities $A, B, C$ are [1rd mutually exclusive and exhaustive], then the probabilities $\\mathbb P(A) + \\mathbb P(B) + \\mathbb P(C)$ should sum to $1.$ If there's no further possibilities $d,$ we can convert the relative odds $(a : b : c)$ into the probabilities $(\\frac{a}{a + b + c} : \\frac{b}{a + b + c} : \\frac{c}{a + b + c}).$ The process of dividing each term by the sum of terms, to turn a set of proportional odds into probabilities that sum to 1, is called [1rk normalization].\n\nWhen there are only two terms $x$ and $y$ in the odds, they can be expressed as a single ratio $\\frac{x}{y}.$ An odds ratio of $\\frac{x}{y}$ refers to odds of $(x : y),$ or, equivalently, odds of $\\left(\\frac{x}{y} : 1\\right).$ Odds of $(x : y)$ are sometimes called odds ratios, where it is understood that the actual ratio is $\\frac{x}{y}.$',
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