|
set difference
|
\[
A-B=AB^c
\]
|
|
symmetric difference.
In probability, symmetric difference means
one and only one of event $A$ and $B$ is realized.
|
\[
A\Delta B=\left(A-B\right)\cup\left(B-A\right)=AB^c\cup BA^c
\]
|
|
set complement
|
\[
A\subset B\Rightarrow B^c\subset A^c
\]
|
|
sets to disjoint sets
|
\[
A\subset B\Rightarrow B=A\cup A^cB
\]
|
|
sets to disjoint sets
|
\[
A\cup B=A\cup A^cB
\]
|
|
sets to disjoint sets
(used in the next chapter)
|
\[
A=AB\cup AB^c
\]
|
|
sets additive identity
|
\[
A\cup\emptyset=A
\]
|
|
sets zero-product rule
|
\[
A\cap\emptyset=\emptyset
\]
|
|
sets identity sample space addition rule
|
\[
A\cup S=S
\]
|
|
sets identity sample space multiplication rule
|
\[
A\cap S=A
\]
|
|
27
|
De Morgan's law
|
\[
\left(A\cup B\right)^c=A^cB^c
\]
|
|
27
|
De Morgan's law
|
\[
\left(AB\right)^c=A^c\cup B^c
\]
|
|
29
|
relative frequency definition of probability
|
\[
\Pr{\left(E\right)}
= \lim_{n \to \infty} \frac{n \left( E \right)}{n}
\]
|
|
29
|
Axiom 1. First axiom of probability
|
\[
0\le\Pr{\left(E\right)}\le1
\]
|
|
29
|
Axiom 2. Second axiom of probability
|
\[
\Pr{\left(S\right)}=1
\]
|
|
30
|
Axiom 3. Third axiom of probability
for mutually exclusive events $E_i$ when $i\geq1.$
|
\[
\Pr{\left(\bigcup_{i=1}^{\infty}E_i\right)}=\sum_{i=1}^{\infty}\Pr{\left(E_i\right)}
\]
|
|
31
|
probability of complement
|
\[
\Pr{\left(A^c\right)}=1-\Pr{\left(A\right)}
\]
|
|
31
|
a theorem
|
\[
A\subset B\Rightarrow\Pr{\left(A\right)}\le\Pr{\left(B\right)}
\]
|
|
32
|
addition rule for probability
|
\[
\Pr{\left(A\cup B\right)}=\Pr{\left(A\right)}+\Pr{\left(B\right)}-\Pr{\left(AB\right)}
\]
|
|
multiplication rule for probability
|
|
|
extended multiplication rule for probability
|
|
Alex Notes