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Probability is the language of uncertainty. In Machine Learning, we almost never say "This image is a cat." Instead, we say "I am 98% sure this image is a cat."
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Bayes' Theorem is the mathematical formula for updating our beliefs when we see new evidence.
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Conditional probability is answering the question: "How does my guess change if I learn a new clue?"
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A distribution is just a map of where data tends to live.