MANE 3332.03
Lecture 6, February 11
Agenda
- Complete Chapter 2 lecture
- New: Two Events Quiz (Assigned: 2/11/25, due 2/13/25)
Handouts
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