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Probability II#

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Important Readings

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  • [Tij12], Chapter 6

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In this section, we leave the main textbook to learn Bayes’ Theorem. This helps provide a foundation for further coursework in Bayesian statistics or just help you interact with nerdy people who talk about “Bayesian updating” in response to new information. Bayes’ Theorem helps solve for a conditional probability \(\mathbb{P}(A \mid B)\) given some other information. This involves “updating” because after observing \(B\), the probability of \(A\) changes from \(\mathbb{P}(A)\) to \(\mathbb{P}(A \mid B)\). The unconditional is a prior belief and the conditional probability is a posterior belief. Bayesian probabilities and statistics are important in any kind of environment where information is gathered. For example, medical doctors use Bayes’ Theorem to assess the chance of a disease (\(A\)) given a particular test result (\(B\)).

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Nothing in this section will require events to be independent.

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Chance Trees#

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A probability tree helps organize information regarding conditional probabilities or almost anything that involves the general multiplication rule.

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Fig. 41 A generic chance tree.#

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And here is something more concrete. For English words of Latin or French origin, the letter Q is always followed by a U. This means if we know that such a word starts with Q, the conditional probability of the second letter being U is 1 (100%). However, if the first letter is not Q, a U isn’t very likely. If second letters are drawn randomly for non-Q words, this probability would be 1/26. We’ll also assume, for simplicity, that there is a 1/26 probability that a word starts with Q.

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Fig. 42 Trees are especially helpful when the probability of a second event (like the letter U) depends on the first event (the previous letter being Q or not Q).#

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