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MANE 3332.04

Lecture 6, February 12

Agenda

  • Complete Chapter 2 lecture
  • New: Two Events Quiz (assigned 2/12/2025, due 2/17/205)
  • Attendance

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

problem 2-76

Independent Events

  • Two events are independent if any one of the following is true:

    1. \(P(A|B)=P(A)\)

    2. \(P(B|A)=P(B)\)

    3. \(P(A\cap B)=P(A)P(B)\)

  • 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