introduction

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

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

Axioms:

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

Consequences:

1. $P(A)le 1$
2. $P(phi)=0$
3. $P(A)+P(A^{C})=1$
4. For mutually disjoint sets $A_1,A_2,ldots,A_k$,
   $P(A_1cup A_2cupldotscup A_k)=P(A_1)+P(A_2)+ldots+P(A_k)$
5. $P(s_1,s_2,ldots,s_k)=P(lbrace s_1rbrace)+P(lbrace s_2rbrace)+ldots+P(lbrace s_krbrace)$
6. If $Asubset B$, $P(A)le P(B)$
7. $P(Acup B)=P(A)+P(B)-P(Acap B)$
8. $P(Acup B)le P(A)+P(B)$
9. $P(Acup Bcup C)=P(A)+P(Bcap A^{C})+P(Ccap B^{C} cap A^{C})$

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

• 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 $frac{1}{16}$.
  • $P(X=1)=4timesfrac{1}{16}=frac{1}{4}$
    Let $Z=min(X,Y)$
  • $P(Z=4)=frac{1}{16}$
  • $P(Z=2)=5timesfrac{1}{16}=frac{5}{16}$

• $(x,y)$ such that $0le x,yle 1$.
  Sample space:
  
• Uniform probability law: Probability = Area
  • $P(lbrace(x,y)|x+yle1/2rbrace)=frac{1}{2}timesfrac{1}{2}timesfrac{1}{2}=frac{1}{8}$

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

If $A_1,A_2,ldots$ is an infinite sequence of disjoint events, then $P(A_1cup A_2cupldots)=sum P(A_i)$. 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 $P(Omega)$
$$
P(Omega)=P(lbrace (x,y)|0le x,yle 1rbrace)=sum P((x,y))=sum 0=0.
$$
which is impossible.

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

• 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: