Probability

Set Notation

• $S$$\large S$: sample space, set of all possible outcomes of an experiment
• $E$$\large E$: event, a subset of the sample space
• $E_1\cup E_2$$\large E_1 \cup E_2$: the union of $E_1$$\large E_1$ and $E_2$$\large E_2$
• $E_1{E}_2=E_1\cap E_2$$\large E_1 E_2 = E_1 \cap E_2$: the intersection of $E_1$$\large E_1$ and $E_2$$\large E_2$
• $\bigcup _{n=1}^{\mathrm{\infty }}{E}_n=E_1\cup E_2\cup E_3...$$\large \bigcup\limits^\infty_{n = 1}E_n = E_1 \cup E_2 \cup E_3 ...$
• $\bigcap _{n=1}^{\mathrm{\infty }}{E}_n=E_1\cap E_2\cap E_3...$$\large \bigcap\limits^\infty_{n = 1}E_n = E_1 \cap E_2 \cap E_3 ...$
• $E^C$$\large E^C$: all outcomes not in $E$$\large E$
• $E_1\subset E_2$$\large E_1 \subset E_2$: implies all elements in $E_1$$\large E_1$ are contained by $E_2$$\large E_2$
• $E_1\subset E_2$$\large E_1 \subset E_2$ and $E_2\subset E_1⟹E_1=E_2$$\large E_2 \subset E_1 \implies E_1 = E_2$

De Morgan's Laws

Independent Events

Independent events are events such that their outcomes are independent of one another.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time.

Probability Axioms

Axiom 1

The probability of an event must be a real-valued number between 0 and 1 inclusive.

Axiom 2

The probability of the entire sample space is 1.

Axiom 3

For any sequence of mutually exclusive events $E_1,E_2,...,E_n$$\large E_1, E_2, ..., E_n$, the probability of the union of these events is the sum of the probability of these events.

Other Propositions

• $P(E^c)=1-P(E)$$\Large P(E^c) = 1 - P(E)$
• $E_1\subset E_2⟹P(E)\le P(F)$$\Large E_1 \subset E_2 \implies P(E) \le P(F)$

Inclusion-Exclusion Principle

Conditional Probability

Conditional probability refers to the probability of an event occurring given the fact that another event has occurred.

Bayes' Theorem

Consider mutually exclusive events $E_1,E_2,E_3,...,E_n$$\large E_1, E_2, E_3, ..., E_n$, such that $\bigcup _{i=1}^n{E}_i=S$$\large \bigcup\limits^n_{i = 1} E_i = S$. Then,