Lecture - Probability MT22, III

Created: October 17, 2022 | About these notes | View in context | Study these flashcards

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What does it mean for a set $S$ to be countable?

Either there exists a bijection $\mathbb{N} \to S$ or $S$ is finite.

What’s the formula for the conditional probability $\mathbb{P}(A \vert B)$?
\[\frac{\mathbb{P}(A \cap B)}{P(B)}\]

What’s the formula for the probability of intersections $\mathbb{P}(A _ 1 \cap A _ 2 \cap \ldots \cap A _ n)$ in terms of condition probability?
\[P(A_1) P(A_2 | A_1)P(A_3 | A_1 \cap A_2) \times \ldots \times P(A_n|A_1 \cap A_2 \cap\ldots \cap A_n)\]

What’s another name for the Law of Total Probability?

The partition theorem.

What does the Law of Total Probability state about a partition of $\Omega$ formed by a family of events $\{B _ 1, B _ 2, \ldots, B _ n\}$ and the probability $\mathbb{P}(A)$ for any $A \in \mathcal{F}$?
\[\mathbb{P}(A) = \sum_{i \ge 1}\mathbb{P}(A|B_i)\mathbb{P}(B_i)\]

What’s a common trick involving proofs about probabilities?

Writing $\mathbb{P}(A) = \mathbb{P}(A \cup \Omega)$

What does Bayes’ theorem for a partition of $\Omega$ formed by a family of events $\{B _ 1, B _ 2, \ldots, B _ n\}$ say about $\mathbb{P}(B _ k \vert A)$?
\[\mathbb{P}(B_k | A) = \frac{\mathbb{P}(A | B_k) \mathbb{P}(B_k)}{\sum_{i\ge1} \mathbb{P}(A | B_i) \mathbb{P}(B_i)}\]


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