#import "@preview/ilm:1.4.1": *


// #show: ilm.with(
//   title: [],
//   author: "Yadunand Prem",
//   table-of-contents: none,
// )
#set text(lang: "en", font: ("SF Pro Display"))
#set heading(numbering: "1.")
#show raw: set text(font: "SF Mono")

= Reference
- Sample Space ($S$): Set of all possible outcomes of an experiment
  - Can vary depending on the problem of interest
- Sample Point: Outcome of sample Space (Element)
- Event: Subset of sample space (Set)

== Set Operations
- $A union B = {x : x in A "or" x in B}$
- $A inter B = {x : x in A "and" x in B}$
- $A' = {x : x in S "and" x in.not A}$
- $limits(union.big)^n_(i=1) A_i = A_1 union A_2 union ... union A_n = {x: x in A_1 "or" " ... or"  x in A_n}$
- Mutually exclusive / disjoint - $A inter B = emptyset$
- Contained - All in A are also elements in B, A is contained in B, $A subset B$ or $B supset A$
- Equivalent - $A subset B$ and $B subset A$, then $A = B$

== Set Operations
- $A inter A' = emptyset$
- $A inter emptyset = emptyset$
- $A union A' = S$
- $(A')' = A$
- $A union (B inter C) = (A union B) inter (A union C)$
- $A inter (B union C) = (A inter B) union (A inter C)$
- $A union B = A union (B inter A')$
- $A = (A inter B)  union (A inter B')$
- $(A_1 union A_2 union ... union A_n)' = A_1^' inter A_2^' inter ... inter A_n^'$
- $(A_1 inter A_2 inter ... inter A_n)' = A_1^' union A_2^' union ... union A_n^'$

== Counting Methods
- Multiplication Principle - $r$ different experiments to be performed sequentially. Then there are $n_1n_2...n_r$ possible outcomes for $r$ experiments
- Addition Principle - experiment can be performed by $k$ different procedures. Suppose ways under different procedures *do not overlap*. Then total is $n_1 + ... + n_k$
- *Permutation* is selection of $r$ objects out of $n$. Order is taken into consideration.
  $ P^n_r = n! / (n-r)! = n(n-1)(n-2)...(n-(r-1)) $ (When $r = n, P^n_n = n!$)
- *Combination* is selection of $r$ objects out of $n$, without regard for order.
  $ vec(n, r) = n!/(r!(n-r!)) = vec(n, n-r) $
  - Intuition: In terms of permutation, no of ways to choose and arrange $r$ objects out fo $n$ is $P_r^n$
  - This can be also done by the following:
    - Select $r$ objects out of $n$ without regard to order: $vec(n, r)$ ways
    - For each combination, permute its $r$ objects: $P^r_r$ ways
    - For each combination, permute its $r$ objects: $P^r_r$ ways
    - $vec(n, r) times P^r_r = P^n_r$

== Probability
- Probability is chance or how likely a certain event may occur. Let $A$ be an event in an experiment. $P(A)$ is to quantify how likely $A$ may occur.
=== Axioms
Probability, $P(dot)$ is a function on the collection of events in the sample space satisfying:
- For any event $A$, $0 lt.eq P(A) lt.eq 1$
- For the sample space $P(S) = 1$
- For any 2 mutually exclusive event $A$ and $B$, that is $A inter B = emptyset$, $P(A union B) = P(A) + P(B)$
- $P(emptyset) = 0$
- $P(A') = 1-P(A)$
- $P(A) = P(A inter B) + P(A inter B')$
- $P(A union B) = P(A) + P(B) - P(A inter B)$
- $A subset B$, then $P(A) lt.eq P(B)$