Lecture Topic: Probability

Experiment: An act where the outcome is subject to uncertainty

Sample space: The set of all possible outcomes of an experiment
Example:
- Flip a coin: $S = \{H, T\}$
- Throw a dice: $S = \{1, 2, 3, 4, 5, 6\}$

Events: An event is a collection (subset) of outcomes contained in the sample space S
$$
S = 
\begin{Bmatrix}
1,1 & 1,2 & 1,3 & 1,4 & 1,5 & 1,6 \\
2,1 & 2,2 & 2,3 & 2,4 & 2,5 & 2,6 \\
3,1 & 3,2 & 3,3 & 3,4 & 3,5 & 3,6 \\
4,1 & 4,2 & 4,3 & 4,4 & 4,5 & 4,6 \\
5,1 & 5,2 & 5,3 & 5,4 & 5,5 & 5,6 \\
6,1 & 6,2 & 6,3 & 6,4 & 6,5 & 6,6
\end{Bmatrix}
$$
A = First and second elements are the same
$A = \{(1,1) \ (2,2) \ (3,3) \ (4,4) \ (5,5) \ (6,6)\}$

Union: Union of two events, A and B, ($A \cup B$)

Compliment: The compliment of an event A, is the set of all outcomes in S that are not in A

Mutually Exclusive Events: When and B have no common outcomes, they are said to be mutually exclusive or disjoint events, i.e. $P=(A \cap B) = 0$ 

Axiom:
1. For event A, P(A) >= 0
2. P(S) = 1
3.  (Need to look at slides to correct)
   (a) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
   (b) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
   Proposition:
   1. For event A, P(A) = 1 - P(A')
   2. If and and B are mutually exclusive then $P(A \cap B) = 0$

Permutation: Any ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects. The number of permutations of size k that can be constructed from n objects is denoted by $P_{k,n}$ and calculated by:
$$
P_{k,n} = \frac{n!}{(n-k)!}
$$

Combination: Given a set of n, any unordered sub-set of k of the objects is called a combination. The set of combinations of size k that can be formed from n distinct objected will be denoted by $C_{k,n}$ and calculated by:
$$
C_{k,n} = \frac{n!}{k!(n-k)!} = \begin{pmatrix}n \\ k \end{pmatrix}
$$

(NEED TO REVIEW THIS)

Example: S = {1, 2, 3, 4, 5}
$A_1$ = {1,2,3}, $A_2$ = {2,3,5}, $A_3$ = {2,3,1}

Permutation:
5! = 120
(5-3)! = 2! = 2
120 / 2 = 60

Combination:
5!/2!3! = 60/6 = 10
0! = 1
5 out of 5
5!/(5-5)! = 120/1 = 120

