MANE 3332.03
Lecture 5, February 6
Agenda
- Course Questions
- Continue Chapter 2 lecture
- Single Event Quiz (Assigned: 2/6/25, due: 2/11/25)
- Two Events Practice Problems (Assigned: 2/6/25, due 2/11/25)
Handouts
Conditional Probability
-
Hayter (2002) states that "For experiments with two or more events of interest, attention is often directed not only at the probabilities of individual events but also at the probability of an event occurring conditional on the knowledge that another event has occurred."
-
The conditional probability of an event \(B\) given an event \(A\), denoted \(P(B|A)\) is
\[
P(B|A)=\frac{P(A\cap B)}{P(A)}
\]
for \(P(A)>0\)
- Consider problems 2-99
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
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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