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Introduction-to-Probability-MIT(01)

LECTURE 1: Probability models and axioms

Two Steps

  1. Describe possible outcomes
  2. Describe beliefs about likelihood of outcomes

Sample Space

Sample space is the set of all outcomes of the experiment. It can be discrete or continuous, finite or infinite.

Event: subset of sample space

Probability Axioms and Derived Consequences

Axioms:

  1. Nonnegativity: $P(A)\ge0$
  2. Normalization: $P(\Omega)=1$
  3. (Finite) additivity: if $A\cap B=\phi$, then $P(A\cup B)=P(A)+P(B)$

Consequences:

  1. $P(A)\le 1$
  2. $P(\phi)=0$
  3. $P(A)+P(A^{C})=1$
  4. For mutually disjoint sets $A_1,A_2,\ldots,A_k$,
    $P(A_1\cup A_2\cup\ldots\cup A_k)=P(A_1)+P(A_2)+\ldots+P(A_k)$
  5. $P(s_1,s_2,\ldots,s_k)=P(\lbrace s_1\rbrace)+P(\lbrace s_2\rbrace)+\ldots+P(\lbrace s_k\rbrace)$
  6. If $A\subset B$, $P(A)\le P(B)$
  7. $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
  8. $P(A\cup B)\le P(A)+P(B)$
  9. $P(A\cup B\cup C)=P(A)+P(B\cap A^{C})+P(C\cap B^{C} \cap A^{C})$

Probability Calculation

Four Steps

  1. Specify the sample sapce
  2. Specify a probability law
  3. Identify an event of interest
  4. Calculate

Discrete and Finite

  • Two rolls of a tetrahedral die. X for the points of the first roll. Y for the points of the second roll. Sample Space:
  • Discrete uniform probability law: every outcome has the same probability $\frac{1}{16}$.
    • $P(X=1)=4\times\frac{1}{16}=\frac{1}{4}$
      Let $Z=\min(X,Y)$
    • $P(Z=4)=\frac{1}{16}$
    • $P(Z=2)=5\times\frac{1}{16}=\frac{5}{16}$

Continuous

  • $(x,y)$ such that $0\le x,y\le 1$.
    Sample space:
  • Uniform probability law: Probability = Area
    • $P(\lbrace(x,y)|x+y\le1/2\rbrace)=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1}{8}$

Discrete and Infinite

  • Tossing a coin, record the number of times until its head faces up.
    Sample space: $\lbrace 1,2,\ldots\rbrace$
    Probability:
  • $P(n)=\frac{1}{2^{n}}$
    $P(\text{outcome is even})=P(2)+P(4)+\ldots=\frac{1}{4}\frac{1}{1-\frac{1}{4}}=\frac{1}{3}$

Countable additivity axiom

If $A_1,A_2,\ldots$ is an infinite sequence of disjoint events, then $P(A_1\cup A_2\cup\ldots)=\sum P(A_i)$. It is this axiom that supports the calculation in Discrete and Infinite section. If it is not countable, consider the section Continuous and try to calculate $P(\Omega)$
$$
P(\Omega)=P(\lbrace (x,y)|0\le x,y\le 1\rbrace)=\sum P((x,y))=\sum 0=0.
$$
which is impossible.

Intepretations of probability theory

  1. (Narrow) a branch of math: Axioms $\Rightarrow$ Theorems
  2. (Objective) Probability = Frequencies in infinite number of experiments
  3. (Subjective) Beliefs or Preferences

Role of probability theory

  • A systematic way of analyzing phenomena with uncertain outcomes
  • Whether the probability is useful for making predictions and decisions or not is related to whether the model fits the reality well or not.
    • Statistics is to use data from real world to come up with good models for probability theory.

Relation: