Probability

A probability assignment $P$ for a random experiment is a numerically valued function that assigns $P(A)$ to every event $A$ such that the following axioms are satisfied

1. $P(A)\ge 0$ for every event A
2. $P(S)=1$
3. If $A_1,A_2,\dots$ is a sequence of mutually exclusive events, then $P(\bigcup _{i=1}^{\infty }{A}_i)=\sum _{i=1}^{\infty }P(A_i)$
   • Note: also applies to finite sets because you can append an infinite number of $\varnothing$ and it remains mutually exclusive

Terminology

• Sample Space
  • The set of all possible outcomes of the experiments listed in a mutually exclusive and exhaustive way
  • Can be discrete (countable) or continuous (uncountable)
  • Notated $S$
• Events
  • Any collection of sample points
  • A subset of the sample space
  • $\mathcal{A}$ = collection of possible events = power set of $S$

Laws

• $P(\varnothing )=0$

• $P(A\cup B)=P(A)+P(B)\text{ if }A\cap B=\varnothing$

• $P(\overline{A})=1-P(A)$

• If $A\subset B$, then $P(A)\le P(B)$

• Multiplicative Law: $P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)$

• Inclusion Exclusion Law: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

  • General: $P\left(\bigcup _{i=1}^k{A}_i\right)=\sum _{i=1}^{k}P(A_i)-\sum _{\text{unordered pairs }{i}_1,i_2}P(A_{i_1}\cap A_{i_2})+\sum _{\text{unordered pairs }{i}_1,i_2,i_3}P(A_{i_1}\cap A_{i_2}\cap A_{i_3})-\cdots +(-1{)}^{k-1}P\left(\bigcap _{i=1}^k{A}_i\right)$

Counting Rules

1. If there is an experiment where all outcomes are equally likely, we assign probability to each sample point $P(A)=\frac{|A|}{|S|}$

Finding count: Counting

Conditional Probability Random Variables