Notes - Probability MT22, Probability spaces

In probability, what does $\Omega$ represent?

The set of all possible outcomes, i.e. the sample space.

What’s the sample space for a single coin flip?

\[\\{H, T\\}\]

What is an event, in terms of the sample space $\Omega$?

A subset of the sample space.

If the observed outcome of an experiment is $\omega$, $\omega \in \Omega$, what does it mean for event A to occur?

\[\omega \in A\]

In the case where $\Omega$ is finite and all outcomes are equally likely, what’s the formula for the probability of $A$?

\[\frac{ \vert A \vert }{ \vert \Omega \vert }\]

What’s the formal definition of a probability space?

A triplet $(\Omega, \mathcal{F}, \mathbb{P})$.

What’s $\mathcal{F}$ in a probability space?

A set of events, subsets of $\Omega$.

What two sets does the function $\mathbb{P}$ map between?

\[\mathcal{F} \to \mathbb{R}\]

What’s a common trick involving proofs about probabilities?

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

How many axioms in probability are there relevant to $\mathcal{F}$?

3

What’s the axiom relevant to $\mathcal{F}$ in probability about the overall sample space, $\Omega$?

\[\Omega \in \mathcal{F}\]

What’s the axiom relevant to $\mathcal{F}$ in probability about an event $A$ and its complement $A^C$ ?

\[A\in \mathcal{F} \implies A^C \in \mathcal{F}\]

What’s the axiom relevant to $\mathcal{F}$ in probability about the union of events?

\[A, B \in \mathcal{F} \implies A \cup B \in \mathcal{F}\]

How many axioms are there relevant to $\mathbb{P}$ are there in probability?

3

What’s the axiom relevant to $\mathbb{P}$ in probability about non-negativity?

\[\forall A \in \mathcal{F} , \mathbb{P}(A) \ge 0\]

What’s the axiom relevant to $\mathbb{P}$ in probability about the sample space $\Omega$?

\[P(\Omega) = 1\]

What’s the axiom relevant to $\mathbb{P}$ in probability about disjoint events $A$ and $B$?

\[P(A \cup B) = P(A) + P(B)\]

@State the three axioms related to $\mathcal F$ in a probability space $(\Omega, \mathcal F, \mathbb P)$.

$\Omega \in \mathcal F$,
If $A \in \mathcal F$, then $A^C \in \mathcal F$,
If $\{A _ i : i \in I\}$ is a finite or countably infinite collection of members of $\mathcal F$, then $\cup _ {i \in I} A _ i \in \mathcal F$.

@State the three axioms related to $\mathbb P$ in a probability space $(\Omega, \mathcal F, \mathbb P)$.

$\mathbb P(A) \ge 0$ for all $A \in \mathcal F$.
$\mathbb P (\Omega) = 1$
If $\{A _ i : i \in I\}$ is a finite or countably infinite collection of members of $\mathcal F$, and they are pairwise disjoint, then $\cup _ {i \in I} A _ i = \sum _ {i \in I} \mathbb P (A _ i)$.

Probability MT22, Probability spaces