Probability MT22, Basic definitions

Created: October 21, 2025 | Updated: October 21, 2025 | About these notes | View in context | Study these flashcards

Flashcards

Given an event $B$, @define the “odds” of $B$.
\[\frac{\mathbb{P}(B)}{\mathbb{P}(B^c)}\]

Given an event $A$, @define the “conditional odds” of $A$ given $B$?
\[\frac{\mathbb{P}(A \vert B)}{\mathbb{P}(A \vert B^c)}\]

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

What’s the formula for the probability of intersections $\mathbb{P}(A _ 1 \cap A _ 2 \cap \ldots \cap A _ n)$ in terms of conditional probability?
\[\mathbb P(A _ 1) \mathbb P(A _ 2 \vert A _ 1)\mathbb P(A _ 3 \vert A _ 1 \cap A _ 2) \times \ldots \times \mathbb P(A _ n \vert A _ 1 \cap A _ 2 \cap\ldots \cap A _ n)\]

What does it mean for a family of events $\{ A _ i, i \in I \}$ to be independent?

For all finite subsets $J$ of $I$

\[\mathbb{P}\left(\bigcup _ {i \in J} A _ i \right) = \prod _ {i \in J} \mathbb{P(A _ i)}\]

What is a useful fact about a family of independent events $A _ 1, A _ 2, \ldots, A _ n$?

You can take the complement of some of them and they are still independent.

What does it mean if $(A _ n) _ {n\ge1}$ is an increasing family of events?
\[A _ n \subseteq A _ {n+1}\]

If $(A _ n) _ {n \ge 1}$ is an increasing family of events, what is the expression for $\lim _ {n \to \infty} \mathbb{P}(A _ n)$ when the random variable is discrete?
\[\mathbb{P}\left(\bigcup _ {n=1}^\infty A _ n\right)\]

If $(A _ n) _ {n \ge 1}$ is an increasing family of events, what is the expression for $\mathbb{P}\left(\bigcup _ {n=1}^\infty A _ n\right)$ when the random variable is discrete?
\[\lim _ {n \to \infty} \mathbb{P}(A _ n)\]

What’s an alternative form of $A^C \cap (B \cup C)$ that might be easier to use in probability formulas?
\[(B \backslash A) \cup (C \backslash A)\]


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