A researcher says that there is a 72% chance a polygraph test (lie detector test

Question

A researcher says that there is a 72% chance a polygraph test (lie detector test) will catch a person who is, in fact, lying. Furthermore, there is approximately a 9% chance that the polygraph will falsely accuse someone of lying (a) If the polygraph indicated that 31% of the questions were answered with lies, what would you estimate for the actual percentage of lies in the answers? Hint: Let B = event detector indicates a lie, we are given P(B) = 0.31. Let A-event person is lying, so Ac-event person is not lying. Then P(B) = P(A and B) + P(AC and B) P(B) = P(A)P(8 | A) + P(Ac)P(B | Ac) Replacing PIAS by 1- P(A) gives P(8) = P(A) , p(8 | A) + [1-P(A)) , p(B | Ac) Substitute known values for P(B), P(B | A), and P(B I Ac) into the last equation and solve for P(A). (Round your answer to two decimal places.) P(A) b If the polygraph indicated that 69% of the questions were answered with les, what would you es mate for the actual percentage oflies Round your answer to one decima place. Need Help?Read It

Explanation / Answer

Here we are given that there is a 72% chance that the detector indicates a lie given that the person is lying. This means that:

P(B | A) = 0.72, therefore P(Bc | A) = 1 - 0.72 = 0.28

Also there is a 9% chance that the detector will falsely accuse someone of lying. Therefore we get:

P(B | Ac) = 0.09 and therefore P(Bc | Ac) = 1 - 0.09 = 0.91

a) Now here we are given that 31% of the people were detected to lie. That means that P(B) = 0.31

Now let the probability that a person really lies be K. That means P(A) = K and hence P(Ac) = 1- K

Now using the law of total probability we get:

P(B) = P(B | A)P(A) + P(B | Ac)P(Ac)

Putting all the given values we get:

0.31 = 0.72K + 0.09*(1-K)

0.31 = 0.72K + 0.09 - 0.09K

0.31 = 0.63K + 0.09

0.63K = 0.31 - 0.09 = 0.22

Therefore, we get:

K = 0.22/0.63 = 0.35

Therefore we get P(A) = 0.35

b) Now here instead of P(B) = 0.31, we have been given P(B) = 0.69

Again using the same equation:

P(B) = P(B | A)P(A) + P(B | Ac)P(Ac)

0.69 = 0.72K + 0.09*(1-K)

0.69 = 0.72K + 0.09 - 0.09K

0.63K = 0.6

K = 0.6/0.63 = 0.95 that would be rounded to 1.0 ( rounded to 1 decimal place )

Therefore P(A) = 1 here. ( rounded to 1 decimal place )

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