Lecture - Probability MT22, II

What’s the stars and bars method?

Solving combinatorial problems involving bins by considering the permutations of stars and bars.

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?

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

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 $B$?

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