Question 1197221: A truth serum given to a suspect is known to be 90 % reliable when the person is guilty and 99 % reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Think of an infection rates problem such as this one
https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1196062.html
I'll be following the same type of outline but with a few slight modifications.
positive test = serum indicates the person is guilty
negative test = serum indicates the person is innocent
Consider a town of 10,000 people.
5% of this town has committed a crime
5% of 10,000 = 0.05*10,000 = 500 people committed a crime
Of the 500 people who committed a crime, the truth serum will return a guilty verdict correctly 90% of the time
90% of 500 = 0.90*500 = 450 people test positive
We consider this the true positives.
The remaining 500-450 = 50 people who committed a crime get "not guilty" verdicts incorrectly. These are false negatives.
Since this town has 10,000 people and 500 committed a crime, that leaves 10,000-500 = 9500 people who didn't commit a crime.
The test is 99% reliable when the person is innocent.
Meaning 0.99*9500 = 9405 truly innocent people will get a correct "not guilty" verdict. These are the true negatives.
The remaining 9500-9405 = 95 people get a false positive. The serum says guilty, but the person is actually innocent.
Here's a chart to help remember the various terms discussed.
| Positive Test | Negative Test | | Crime | True positive | False negative | | No Crime | False positive | True negative |
And here's the chart of values for this specific problem. Each value was calculated in previous paragraphs.
| Positive Test | Negative Test | Total | | Crime | 450 | 50 | 500 | | No Crime | 95 | 9405 | 9500 | | Total | 545 | 9455 | 10,000 |
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With all that set up in mind, we're then asked this question:
"If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?"
In other words, we're given the serum states "guilty".
So we can narrow our focus on just the "positive test" cases only.
There are 450 correct guilty verdicts (true positive) and 95 incorrect guilty verdicts (false positive)
450+95 = 545 guilty verdicts in all
The probability of the person being innocent is 95/545 = 0.1743 = 17.43% approximately
Round the decimal and percentage values however your teacher instructs.
Side note: The fraction 95/545 fully reduces to 19/109
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