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| + | [[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 2: The False Positive Paradox == | == Example 2: The False Positive Paradox == | ||
| + | by [[User:Mhossain|Maliha Hossain]], proud Member of [[Math_squad | the Math Squad]]. | ||
| + | ---- | ||
| + | <pre>keyword: probability, false positive, Bayes' Theorem, Bayes' Rule </pre> | ||
| + | |||
The false positive paradox occurs when false positive tests are more probable than true positive tests. The fewer the number of incidents in the overall population, the higher the likelihood of a false positive test. | The false positive paradox occurs when false positive tests are more probable than true positive tests. The fewer the number of incidents in the overall population, the higher the likelihood of a false positive test. | ||
| − | The following example | + | The following example employs Bayes' Theorem to illustrate how a high false positive rate can affect the results of a drug test. |
A manufacturer claims that its product can detect drug use among athletes 97% of the time (i.e. the test will show a positive 97% of the time given that the athletes used drugs). However, there is a 10% chance of a false alarm (i.e. non drug users will show positive results 10% of the time). Given that only 5% of the team actually use drugs, what is the probability that an athlete who tested positive is a non user? | A manufacturer claims that its product can detect drug use among athletes 97% of the time (i.e. the test will show a positive 97% of the time given that the athletes used drugs). However, there is a 10% chance of a false alarm (i.e. non drug users will show positive results 10% of the time). Given that only 5% of the team actually use drugs, what is the probability that an athlete who tested positive is a non user? | ||
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By Bayes' Theorem, we have that | By Bayes' Theorem, we have that | ||
| − | <math>P[D'|Y] = \frac{P[Y|D']P[D']}{P[Y|D]P[D] + P[Y|D']P[D']} | + | <math> |
| − | + | \begin{align} | |
| − | + | P[D'|Y] &= \frac{P[Y|D']P[D']}{P[Y|D]P[D] + P[Y|D']P[D']} \\ | |
| + | &= \frac{0.01\times 0.95}{0.97 \times 0.05 + 0.1 \times 0.95} \\ | ||
| + | &= 0.662 | ||
| + | \end{align} | ||
| + | </math> | ||
So we see that a positive test is more likely to be a false alarm since the false positive rate is too high for this low incidence population. | So we see that a positive test is more likely to be a false alarm since the false positive rate is too high for this low incidence population. | ||
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* Comment / question 1 | * Comment / question 1 | ||
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---- | ---- | ||
| + | |||
| + | [[Bayes_theorem_S13|Back to Tutorial]] | ||
[[Math_squad|Back to Math Squad page]] | [[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 13:09, 25 November 2013
- ↳ Bayes' Theorem
- ↳ Example 1: Quality Control
- ↳ Example 2: False Positives
- ↳ Example 3: Monty Hall Problem
Example 2: The False Positive Paradox
by Maliha Hossain, proud Member of the Math Squad.
keyword: probability, false positive, Bayes' Theorem, Bayes' Rule
The false positive paradox occurs when false positive tests are more probable than true positive tests. The fewer the number of incidents in the overall population, the higher the likelihood of a false positive test.
The following example employs Bayes' Theorem to illustrate how a high false positive rate can affect the results of a drug test.
A manufacturer claims that its product can detect drug use among athletes 97% of the time (i.e. the test will show a positive 97% of the time given that the athletes used drugs). However, there is a 10% chance of a false alarm (i.e. non drug users will show positive results 10% of the time). Given that only 5% of the team actually use drugs, what is the probability that an athlete who tested positive is a non user?
Let $ D $ be the event that the athlete used drugs.
Therefore, $ D' $ is the event that the athlete did not use drugs.
Let $ Y $ be the event that the test result was positive
Therefore a negative result is described by the event $ Y' $
From the problem statement, we can infer the following.
$ P[D] = 0.05 $
$ P[D'] = 1 - P[D'] = 0.95 $
$ P[Y|D] = 0.97 $ (i.e. the probability of a positive test given the athlete used drugs)
$ P[Y'|D] = 1 - P[Y|D] = 0.03 $ (i.e. the probability of a negative test given the athlete used drugs)
$ P[Y|D'] = 0.1 $ (i.e. the probability that the test was positive given the athlete did not take drugs)
$ P[Y'|D'] = 1 - P[Y|D'] = 0.9 $ (i.e. the probability of a negative test given the athlete did not use drugs)
Now we need to find $ P[D'|Y] $, in other words, the probability that the athlete did not use drugs given the test was positive.
By Bayes' Theorem, we have that
$ \begin{align} P[D'|Y] &= \frac{P[Y|D']P[D']}{P[Y|D]P[D] + P[Y|D']P[D']} \\ &= \frac{0.01\times 0.95}{0.97 \times 0.05 + 0.1 \times 0.95} \\ &= 0.662 \end{align} $
So we see that a positive test is more likely to be a false alarm since the false positive rate is too high for this low incidence population.
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.
