MANE 3332.05
Lecture 5
Agenda
- Continue Chapter 2 Lecture
- Start with Two Events Practice Problems
- Single Event Quiz (assigned 9/11/2025, due 9/15/2025)
- Two Events Practice Problems (assigned 9/16/2025, due 9/18/2025)
Handouts
Two Events Practice Problems
Multiplication Rules
- This rule provides another method for calculating \(P(A\cap B)\) $$ \begin{aligned} P(A\cap B)&=&P(A|B)P(B)=P(B|A)P(A) \end{aligned} $$
- This leads to the total probability rule $$ \begin{aligned} P(B)&=&P(B\cap A)+P(B\cap A^{\prime})\ &=&P(B|A)P(A)+P(B|A^{\prime})P(A^{\prime})\ \end{aligned} $$
- Consider problems from 3rd edition (next slide) and 2-129
Example Problem 2-76
Independent Events
-
Two events are independent if any one of the following is true:
-
\(P(A|B)=P(A)\)
-
\(P(B|A)=P(B)\)
-
\(P(A\cap B)=P(A)P(B)\)
-
-
Consider problem 2-146
Reliability Analysis
-
Reliability is the application of statistics and probability to determine the probability that a system will be working properly
-
Components can be arranged in series. All components must work for the system to work.
\[
P(\mbox{system works})=P(A\mbox{ works})P(B\mbox{ works})
\]
- Components can be arranged in parallel. As long as one component works, the system works.
\[
P(\mbox{system works})=1-(1-P(A\mbox{ works}))\times 1-P(B\mbox{ works}))
\]
- Consider problem 2-157