Lecture - Probability MT22, III

Flashcards

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

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

What’s the formula for the conditional probability $P (A | B)$ ?

\frac{P (A \cap B)}{P (B)}

What’s the formula for the probability of intersections $P (A_{1} \cap A_{2} \cap \dots \cap A_{n})$ in terms of condition probability?

P (A_{1}) P (A_{2} | A_{1}) P (A_{3} | A_{1} \cap A_{2}) \times \dots \times P (A_{n} | A_{1} \cap A_{2} \cap \dots \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 $Ω$ formed by a family of events ${B_{1}, B_{2}, \dots, B_{n}}$ and the probability $P (A)$ for any $A \in F$ ?

P (A) = \sum_{i \geq 1} P (A | B_{i}) P (B_{i})

What’s a common trick involving proofs about probabilities?

Writing $P (A) = P (A \cup Ω)$

What does Bayes’ theorem for a partition of $Ω$ formed by a family of events ${B_{1}, B_{2}, \dots, B_{n}}$ say about $P (B_{k} | A)$ ?

P (B_{k} | A) = \frac{P (A | B_{k}) P (B_{k})}{\sum_{i \geq 1} P (A | B_{i}) P (B_{i})}

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