# Thread: Polygraph test - simple probability with percentages

1. ## Polygraph test - simple probability with percentages

Here's a simple problem from a prep book I'm using to get ready for a test.

A polygraph test is 90% effective in determining if someone is lying. However, it has a 10% error-rate for "false" detection. A congressman who is known to lie 25% of the time is tested and found to be telling the truth.

A. What is the probability that he could still be lying? (note that the correct answer here is supposed to be 3.6%, a number I cannot derive).

B. If the polygraph test says he is lying, what is the probability that he could actually be telling the truth?

2. Originally Posted by MathGeezer
I originally posted this question in the "pre university" section, but got no responses. So, I'm duplicating the post here.

A polygraph test is 90% effective in determining if someone is lying. However, it has a 10% error-rate for "false" detection. A congressman who is known to lie 25% of the time is tested and found to be telling the truth.

A. What is the probability that he could still be lying?

B. If the polygraph test says he is lying, what is the probability that he could actually be telling the truth?
Bayes' theorem:

A.

$\displaystyle P(lying|poly\ true)=\frac{P(poly\ true|lying)P(lying)}{P(poly\ true)}$

You are told:

$\displaystyle P(poly\ true|lying)=0.1$

$\displaystyle P(lying)=0.25$

$\displaystyle P(poly\ true|not\ lying)=0.9$

$\displaystyle P(poly\ true)=P(lying)P(poly\ true|lying)+(1-P(lying))P(poly\ true|not\ lying)$

CB

3. Originally Posted by MathGeezer
Here's a simple problem from a prep book I'm using to get ready for a test.

A polygraph test is 90% effective in determining if someone is lying. However, it has a 10% error-rate for "false" detection. A congressman who is known to lie 25% of the time is tested and found to be telling the truth.

A. What is the probability that he could still be lying? (note that the correct answer here is supposed to be 3.6%, a number I cannot derive).
This is a conditional probability that he was found telling the truth despite him being a liar. This is an example of Bayes' theorem.

$\displaystyle P(A|B)=\frac{P(B|A)P(A)}{P(B)}$

Let A be him lying and B be him passing.

$\displaystyle P(Liar|Passed) = \frac{P(Passed|Liar)P(Liar)}{P(Passed)}$

• We know that the congressman is a liar 25% of the time - $\displaystyle P(Liar) = 0.25$.
• We also know that the the probability the machine passes someone who is a liar (i.e. fails) is 10%.
• The probability of the machine passing him in general should be 90% of the time for when he is not a liar (75%) and 10% of the time when he is a liar (25%). The weighted average of that is $\displaystyle P(passed)=0.9\cdot0.75+0.1\cdot0.25=0.675+0.025=0. 7$

$\displaystyle P(Liar|Passed)=\frac{0.25\cdot0.1}{0.7}=0.03571 \approx 0.036$

I will let you take care of the 2nd part.

4. Originally Posted by MathGeezer
Here's a simple problem from a prep book I'm using to get ready for a test.

A polygraph test is 90% effective in determining if someone is lying. However, it has a 10% error-rate for "false" detection. A congressman who is known to lie 25% of the time is tested and found to be telling the truth.
I don't believe it! A congressman who only lies 25% of the time? Impossible!

A. What is the probability that he could still be lying? (note that the correct answer here is supposed to be 3.6%, a number I cannot derive).

B. If the polygraph test says he is lying, what is the probability that he could actually be telling the truth?

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### probability tree with polygraph

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