Question

suppose a police detective is about to administer a polygraph (lie detector) test on an "individual of interest" . when a polygraph is administered to an individual who is lying, the chance that the test detects that the individual is lying is 90%. additionally, when polygraph test is administered to an individual who is telling the truth, the chance that the test indicates that the individual is not lying is 84%. finally, the police detective knows that "individuals of interest" don't tell the truth 10% of the time.

- let L be the event that the person is not telling the truth ( lying).

-let Lbar be the event that the person is telling the truth.

- Let (+) be the event that the polygraph indicates that the individual is not telling the truth

-let (-)be the event that the polygraph indicates that the individual is telling the truth.

(a)- what is the P(-)?

(b)- given that the polygraph indicates that an individual is not telling the truth, what is the probability that he or she is lying?

Answer 1

When a polygraph is administered to an individual who is lying, the chance that the test detects that the individual is lying is 90%

P(+|L) = 90/100 = 0.90

P(-|L) = 1 - 0.90 = 0.10

when polygraph test is administered to an individual who is telling the truth, the chance that the test indicates that the individual is not lying is 84%

P(-|Lbar) =84/100 = 0.84

P(+|Lbar) = 1- 0.84 = 0.16

"individuals of interest" don't tell the truth 10% of the time.

P(L) = 10/100 = 0.1

P(Lbar) = 1-0.1 = 0.9

(a) . P(-) = P(L)P(-|L) + P(Lbar)P(-|Lbar) = 0.1 * 0.1 + 0.9*0.84 = 0.766

P(-) = 0.766

(b) given that the polygraph indicates that an individual is not telling the truth, what is the probability that he or she is lying

Probability that he or she is lying (+) given that the polygraph indicates that an individual is not telling the truth(L) = P(L|+)

By Bayes theorem,

P(L)P(+|L) = 0.1 * 0.90 = 0.09

P(L)P(+|L)+P(Lbar)P(+|Lbar) = 0.1*0.90 + 0.9*0.16 = 0.09+0.144=0.234

Probability that he or she is lying given that the polygraph indicates that an individual is not telling the truth = 0.384615