Check point 1: Basic Set Theory and Probability Definition

Definitions

  • Random Experiment: a phenomenon with an unknown outcome

    • Exp: rolling a dice not knowing what side would land on

  • Element: a possible outcome of an experiment

    • Exp: rolling a 2

  • Sample Space (S): S of an experiment is the set of all possible outcomes of that experiment

    • Exp: S = {1, 2, 3, 4, 5, 6}

  • Event (E): a set containing possible outcomes of an experiment. aka E is a subset of S, denoted E⊆\subseteq S

    • Exp: Event of rolling prime number = {2,3,5}

  • E Complement: Ec (or E’, ~E, E‾\overline{E} ) the complement of E

    • Math: Sample Space (S) - Event (E)

    • Exp: Event of rolling non even number = {1, 3, 5}

  • Union (AND): Union of events is a set consisting of all outcome found in any of the events. In simple words, everything the two or more events cover

    • Exp: A∪\cup B

  • Intersection (OR): Intersection of events is a set consisting of only the common outcome found in any of the events. In simple words, only the common area that the two or more events cover

    • Exp: A∩\cap B

    • Math: : A∩\cap B∩\cap C = A x B x C

  • Mutually Exclusive (Disjoint): if there are no common outcomes, denoted E∩\cap F = 0

    • For 3 or more, every events must be disjoint, not just two

  • Bayes’ Formula = P(B∣A)=P(A∣B∣)P(B)P(A)P\left(B\left|A\right.\right)=\frac{P\left(A\left|B\right|\right)P\left(B\right)}{P\left(A\right)}



Independence

  • Check for independent =

    • P(A∣B)=P(A)P\left(A\vert B\right)=P\left(A\right)

    • P(A∩B)=P(A)×P(B)P\left(A\cap B\right)=P\left(A\right)\times P\left(B\right)

  • Complement

    • if P(A) and P(B) are independent

      • P(A) and P(Bc) are also independent

      • P(Ac) and P(B) are also independent

      • P(Ac) and P(Bc) are also independent



Conditional Probability

  • P(A | B) = A∩BB\frac{A\cap B}{B}


Formula

  • A∪B=A+B−A∩BA\cup B=A+B-A\cap B

  • A∩Bc=A−A∩BA\cap B^{c}=A-A\cap B

  • AC∪B=AC+B−AC∩BA^{C}\cup B^{}=A^{C}+B-A^{C}\cap B

  • (A∩B)C=1−(A∩B)\left(A\cap B\right)^{C}=1-\left(A\cap B\right)^{}

  • (A∪B)C=1−(A∪B)\left(A\cup B\right)^{C}=1-\left(A\cup B\right)^{}

  • P(∅) = 0 (P for mutually exclusive event)


Laws

  • Demorgon’s Law


  • Law of Total Probability

    • probability of an event can be calculated by considering all the different ways it could happen

    • Exp: P(B) = P(B ∩\cap A) +P(B ∩\cap Ac)


  • Probability Axiom



Counting and Combination

  • Counting how many ways can smth happened

    • # of Stage times all the possibility of each stage

      • *the possibility for each stage might not be the same

  • Permutation

    • Definition

      • An ordered arragement is called a Permutation

      • Simple scenario: pick a ball from the pool and pick another way wo putting the previous one back to the pool

    • Formula

      • No repetition allowed Version: n!=n(n−1)(n−2)…3⋅2⋅1n!=n\left(n-1\right)\left(n-2\right)\ldots3\cdot2\cdot1

      • nPrnPr :

        • n = # of all the possibility for the first stage

        • r = how many stage we looking for

      • Repetition

        • we dont minus one of the possibility every time we multiply since we are allowed to repeat the options in each stage.

  • Combination

    • Definition

      • An non-ordered arragement is called a Combination

      • We say N Choose K

        • To find how many combinations of 5 poker cards from a deck we can say 52 Choose 5

    • Formula

      • nCknCk = n!(n−k)!k!\frac{n!}{\left(n-k\right)!k!} :

        • n = # of all the possibility for the first stage

        • r = how many stage we looking for

      • Relationship

    • Example

      • How many combination of friends i can invite to a party if i have 8 but can only invite 2, and 2 of them are not coming if they have to come tgh