Notes - Probability MT22, Basic definitions

**- Course - Probability^U

Flashcards

Odds

@Define the “odds” of an event $B$.

odds-of-an-event

\[\frac{\mathbb{P}(B)}{\mathbb{P}(B^c)}\]

@Define the “conditional odds” of an event $A$ given $B$.

conditional-odds

\[\frac{\mathbb{P}(A \vert B)}{\mathbb{P}(A \vert B^c)}\]

Conditional probability

@Define the conditional probability $\mathbb{P}(A \vert B)$.

conditional-probability

\[\frac{\mathbb{P}(A \cap B)}{\mathbb P(B)}\]

@State the formula for the probability of intersections $\mathbb{P}(A _ 1 \cap A _ 2 \cap \ldots \cap A _ n)$ in terms of conditional probabilities.

probability-chain-rule

\[\mathbb P(A _ 1) \mathbb P(A _ 2 \vert A _ 1)\mathbb P(A _ 3 \vert A _ 1 \cap A _ 2) \times \ldots \times \mathbb P(A _ n \vert A _ 1 \cap A _ 2 \cap\ldots \cap A _ {n-1})\]

Independence

@Define what it means for a family of events $\{ A _ i, i \in I \}$ to be independent.

independent-family-of-events

For all finite subsets $J$ of $I$

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

If $A _ 1, A _ 2, \ldots, A _ n$ are independent events, what is still true after replacing some of them by their complements?

independence-complements

They remain independent.

Increasing families of events

@Define what it means for $(A _ n) _ {n\ge1}$ to be an increasing family of events.

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?

increasing-events-limit-of-probabilities

\[\mathbb{P}\left(\bigcup _ {n=1}^\infty A _ n\right)\]

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?

increasing-events-probability-of-union

\[\lim _ {n \to \infty} \mathbb{P}(A _ n)\]

Set identities

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

complement-intersection-identity

\[(B \backslash A) \cup (C \backslash A)\]