(New page: The following problem has been adapted from a few practice problems from chapter 2 of Probability, Statistics and Random Processes for Electrical Engineers by Alberto Leon-Garcia. The exam...) |
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| − | The following problem has been adapted from a few practice problems from chapter 2 of Probability, Statistics and Random Processes for Electrical Engineers by Alberto Leon-Garcia. The example illustrates how Bayes' Theorem plays a role in quality control. | + | [[Category:math]] |
| + | [[Category:tutorial]] | ||
| + | [[Category:math squad]] | ||
| + | |||
| + | [[Math_squad |Math Squad]] | ||
| + | |||
| + | :↳ [[Bayes_theorem_S13|Bayes' Theorem]] | ||
| + | ::↳ [[Bayes_theorem_eg1_S13|Example 1: Quality Control]] | ||
| + | ::↳ [[Bayes_theorem_eg2_S13|Example 2: False Positives]] | ||
| + | ::↳ [[Bayes_theorem_eg3_S13|Example 3: Monty Hall Problem]] | ||
| + | |||
| + | ---- | ||
| + | |||
| + | == Example 1: Quality Control == | ||
| + | by [[User:Mhossain|Maliha Hossain]], proud Member of [[Math_squad | the Math Squad]]. | ||
| + | ---- | ||
| + | <pre>keyword: probability, Bayes' Theorem, Bayes' Rule </pre> | ||
| + | |||
| + | |||
| + | The following problem has been adapted from a few practice problems from chapter 2 of ''Probability, Statistics and Random Processes for Electrical Engineers'' by Alberto Leon-Garcia. The example illustrates how Bayes' Theorem plays a role in quality control. | ||
A manufacturer produces a mix of "good" chips and "bad" chips. The proportion of good chips whose lifetime exceeds time <math>t</math> seconds decreases exponentially at the rate <math>\alpha</math>. The proportion of bad chips whose lifetime exceeds t decreases much faster at a rate <math>1000\alpha</math>. | A manufacturer produces a mix of "good" chips and "bad" chips. The proportion of good chips whose lifetime exceeds time <math>t</math> seconds decreases exponentially at the rate <math>\alpha</math>. The proportion of bad chips whose lifetime exceeds t decreases much faster at a rate <math>1000\alpha</math>. | ||
| − | Suppose that the fraction of bad chips is <math>p</math>, and of good chips, <math>1 - p</math> | + | Suppose that the fraction of bad chips is <math>p</math>, and of good chips, <math>1 - p</math>. |
Let <math>C</math> be the event that the chip is functioning after <math>t</math> seconds. | Let <math>C</math> be the event that the chip is functioning after <math>t</math> seconds. | ||
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So by the theorem of total probability, we have that | So by the theorem of total probability, we have that | ||
| − | <math>P[C] = P[C|G]P[G] + P[C|B]P[B] | + | <math> |
| − | + | \begin{align} | |
| − | + | P[C] &= P[C|G]P[G] + P[C|B]P[B] \\ | |
| + | &= e^{-\alpha t}(1-p) + e^{-1000\alpha t}p | ||
| + | \end{align} | ||
| + | </math> | ||
Now suppose that in order to weed out the bad chips, every chip is tested for t seconds prior to leaving the factory. the chips that fail are discarded and the remaining chips are sent out to customers. Can you find the value of <math>t</math> for which 99% of the chips sent out to customers are good? | Now suppose that in order to weed out the bad chips, every chip is tested for t seconds prior to leaving the factory. the chips that fail are discarded and the remaining chips are sent out to customers. Can you find the value of <math>t</math> for which 99% of the chips sent out to customers are good? | ||
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The problem requires that we find the value of <math>t</math> such that | The problem requires that we find the value of <math>t</math> such that | ||
| − | <math>P[G|C] = .99</math> | + | <math>P[G|C] = .99 \ </math> |
We find <math>P[G|C]</math> by applying Bayes' Theorem | We find <math>P[G|C]</math> by applying Bayes' Theorem | ||
| − | <math>P[G|C] = \frac{P[C|G]P[G]}{P[C|G]P[G] + P[C|B]P[B]} | + | <math> |
| − | + | \begin{align} | |
| − | + | P[G|C] &= \frac{P[C|G]P[G]}{P[C|G]P[G] + P[C|B]P[B]} \\ | |
| − | + | &= \frac{e^{-\alpha t}(1-p)}{e^{-\alpha t}(1-p) + e^{-1000\alpha t}} \\ | |
| − | + | &= \frac{1}{1 + \frac{pe^{-1000\alpha t}}{e^{-\alpha t}(1-p)}} = .99 | |
| + | \end{align} | ||
| + | </math> | ||
The above equation can be solved for <math>t</math> | The above equation can be solved for <math>t</math> | ||
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<math>t = \frac{1}{999\alpha}ln(\frac{99p}{1-p})</math> | <math>t = \frac{1}{999\alpha}ln(\frac{99p}{1-p})</math> | ||
---- | ---- | ||
| + | |||
| + | == References == | ||
| + | |||
| + | * Alberto Leon-Garcia, ''Probability, Statistics, and Random Processes for Electrical Engineering,'' Third Edition | ||
| + | ---- | ||
| + | |||
| + | ==Questions and comments== | ||
| + | |||
| + | If you have any questions, comments, etc. please post them below: | ||
| + | |||
| + | * Comment / question 1 | ||
| + | |||
| + | ---- | ||
| + | [[Bayes_theorem_S13|Back to tutorial]] | ||
| + | |||
| + | [[Math_squad|Back to Math Squad page]] | ||
| + | |||
| + | |||
| + | <div style="font-family: Verdana, sans-serif; font-size: 14px; text-align: justify; width: 70%; margin: auto; border: 1px solid #aaa; padding: 2em;"> | ||
| + | The Spring 2013 Math Squad 2013 was supported by an anonymous [https://www.projectrhea.org/learning/donate.php gift] to [https://www.projectrhea.org/learning/about_Rhea.php Project Rhea]. If you enjoyed reading these tutorials, please help Rhea "help students learn" with a [https://www.projectrhea.org/learning/donate.php donation] to this project. Your [https://www.projectrhea.org/learning/donate.php contribution] is greatly appreciated. | ||
| + | </div> | ||
Latest revision as of 12:09, 25 November 2013
- ↳ Bayes' Theorem
- ↳ Example 1: Quality Control
- ↳ Example 2: False Positives
- ↳ Example 3: Monty Hall Problem
Example 1: Quality Control
by Maliha Hossain, proud Member of the Math Squad.
keyword: probability, Bayes' Theorem, Bayes' Rule
The following problem has been adapted from a few practice problems from chapter 2 of Probability, Statistics and Random Processes for Electrical Engineers by Alberto Leon-Garcia. The example illustrates how Bayes' Theorem plays a role in quality control.
A manufacturer produces a mix of "good" chips and "bad" chips. The proportion of good chips whose lifetime exceeds time $ t $ seconds decreases exponentially at the rate $ \alpha $. The proportion of bad chips whose lifetime exceeds t decreases much faster at a rate $ 1000\alpha $. Suppose that the fraction of bad chips is $ p $, and of good chips, $ 1 - p $.
Let $ C $ be the event that the chip is functioning after $ t $ seconds. Let $ G $ be the event that the chip is good. Let $ B $ be the event that the chip is bad.
Here's what we can infer from the problem statement thus far:
the probability that the lifetime of a good chip exceeds $ t $: $ P[C|G] = e^{-\alpha t} $
the probability that the lifetime of a bad chip exceeds $ t $: $ P[C|B] = e^{-1000\alpha t} $
So by the theorem of total probability, we have that
$ \begin{align} P[C] &= P[C|G]P[G] + P[C|B]P[B] \\ &= e^{-\alpha t}(1-p) + e^{-1000\alpha t}p \end{align} $
Now suppose that in order to weed out the bad chips, every chip is tested for t seconds prior to leaving the factory. the chips that fail are discarded and the remaining chips are sent out to customers. Can you find the value of $ t $ for which 99% of the chips sent out to customers are good?
The problem requires that we find the value of $ t $ such that
$ P[G|C] = .99 \ $
We find $ P[G|C] $ by applying Bayes' Theorem
$ \begin{align} P[G|C] &= \frac{P[C|G]P[G]}{P[C|G]P[G] + P[C|B]P[B]} \\ &= \frac{e^{-\alpha t}(1-p)}{e^{-\alpha t}(1-p) + e^{-1000\alpha t}} \\ &= \frac{1}{1 + \frac{pe^{-1000\alpha t}}{e^{-\alpha t}(1-p)}} = .99 \end{align} $
The above equation can be solved for $ t $
$ t = \frac{1}{999\alpha}ln(\frac{99p}{1-p}) $
References
- Alberto Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineering, Third Edition
Questions and comments
If you have any questions, comments, etc. please post them below:
- Comment / question 1
The Spring 2013 Math Squad 2013 was supported by an anonymous gift to Project Rhea. If you enjoyed reading these tutorials, please help Rhea "help students learn" with a donation to this project. Your contribution is greatly appreciated.
