A false-positive in a polygraph test, that is, an individual who fails a lying-detector test even through they are telling the truth, has a probability of 15%. Consider the experiment of administering a lie-detector or polygraph test (these are synonyms or words that have the same meaning in common usage) to a group of randomly selected people and recording false-positives. 1) What is the probability that the first false-positive will occur when the third person is tested? 2) What is the probability that fewer than 4 people are tested before the first false-positive occurs? 3) What is the probability that more than 3 people are tested before the first false-positive occurs?

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A false-positive in a polygraph test, that is, an individual who fails a lying-detector test even through they are telling the truth, has a probability of 15%. Consider the experiment of administering a lie-detector or polygraph test (these are synonyms or words that have the same meaning in common usage) to a group of randomly selected people and recording false-positives.  1) What is the probability that the first false-positive will occur when the third person is tested?  2) What is the probability that fewer than 4 people are tested before the first false-positive occurs?  3) What is the probability that more than 3 people are tested before the first false-positive occurs?

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Geometric Distribution: Let there is a sequence of trials, each having two outcomes, say Success and Failure, trials being independent and the probability of success, say p, remaining constant. Then the distribution of the number of failures before the first success follows geometric.Let us assume X be a random variable denoting the number of failures before the first success. Then the probability mass function of X is given by,P(X=x)={(1−p)xp,x=0,1,2,...0otherwise&amp;nbsp;It is denoted by,(First Solution)X∼Geo(p)Let us assume X is a random variable denoting the number of people will be tested before the first false-positive occursThe probability of false-positive is&amp;nbsp;15%=0.15Here,1) each person&#039;s test is independent of the other2) there are only two possibilities in each person&#039;s test viz whether the test result is false-positive or not3) the number of tests before the false-positive can go up to infinityiv) the probability of false-positive remains constant for all the testsIn these circumstances, we can state that,X∼Geo(p)with&amp;nbsp;p=0.15Now, the event that the first false-positive will occur when the third person is tested implies that there will be two persons tested before the first false-positive occurs.Hence, the probability that there will be two persons tested before the first false-positive occurs is&amp;nbsp;P(X=2)=(1−p)2p=(1−0.15)2×0.15=0.1084&amp;nbsp;(rounded to 4 decimal places)Result: The probability that the first false-positive will occur when the third person is tested is 0.1084.(Second Solution)&amp;nbsp;The probability that less than 4 people are tested before the first false-positive occurs isP(X&amp;lt;4)=∑x=03P(X=x)=∑x=03(1−p)xp=p∑x=03(1−p)x=0.15×(1+0.85+0.852+0.853)=0.4780&amp;nbsp;(rounded to 4 decimal places)Answer: The probability that less than 4 people are tested before the first false-positive occurs is 0.4780.(Third Solution)The probability that more than 3 people are tested before the first false-positive occurs isP(X&amp;gt;3)=1−P(X≤3)=1−∑x=03P(X=x)=1−p∑x=03(1−p)x=1−0.15×(1+0.85+0.852+0.853)=1−0.4780&amp;nbsp;(rounded to 4 decimal places)=0.5220Answer: The probability that more than 3 people are tested before the first false-positive occurs is 0.5220.

A false-positive in a polygraph test, that is, an individual who fails

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