Skip to content

Lecture 2

Frequentist Definition of Probability \(\mathbb P=\lim_{n\rightarrow\infty}\frac{n(A)}n\) which \(n(A)\) is the number of time that \(A\) occurred and \(n\) is the number of total experiments

Properties

  1. \(\mathbb P(\emptyset)=0\)
  2. For disjoint events \(A_1,A_2,....,A_n\) that \(\mathbb P(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mathbb P(A_i)\)
  3. For any event \(A\subseteq\Omega\), \(\mathbb P(A^c)=1-\mathbb P(A)\)
  4. For any event \(A,B\subseteq \Omega\) that \(\mathbb P(A)=\mathbb P(A\cap B)+\mathbb (A\cap B^c)\)
  5. For any event \(A\subseteq B\) that \(\mathbb P(B\backslash A)=\mathbb P(B)-\mathbb P(A)\) and \(\mathbb P(A)\leq \mathbb P(B)\)
  6. For any event \(A\subseteq \Omega\) that \(\mathbb P(A)\leq 1\)
  7. For any event \(A,B\subseteq \Omega\) that \(\mathbb P(A\cup B)=\mathbb P(A)+\mathbb (B)-\mathbb P(A\cap B)\)
  8. Property 7 can be extended to more terms
  9. For events \(A_1,...,A_n\subseteq \Omega\) that \(\mathbb P(\bigcup^\infty_{i=1}A_i)\leq \sum^\infty_{i=1}\mathbb P(A_i)\)
  10. For a sequence st \(A_1\subseteq A_2\subseteq A_3...\) then \(\mathbb P(\bigcup^\infty_{i=1}A_i)=\lim_{n\rightarrow\infty}\mathbb P(A_n)\)
  11. For a sequence st \(A_1\supseteq A_2\supseteq A_3...\) then \(\mathbb P(\bigcap^\infty_{i=1}A_i)=\lim_{n\rightarrow\infty} \mathbb P(A_n)\)