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I have been asked to solve this problem:

A policy requiring all hospital employees to take lie detector tests may reduce losses due to theft, but some employees regard such tests as a violation of their rights. To gain some insight into the risks that employees face when taking a lie detector test, suppose that the probability is 0.05 that a lie detector concludes that a person is lying who, in fact, is telling the truth and suppose that any pair of tests are independent.

What is the probability that a machine will conclude that at least one of the three employees is lying when all are telling the truth?

My intuition, which was wrong, lead me to do ($0.95^{2}$)(0.05). I believe that this tells me the probability of two negatives and one positive. However, looking at it again, it looks like this tells me the probability of the first two tests being negative while the third is positive?

The correct answer is 1-($0.95^{3}$).

This confuses me because isn't ($0.95^{3}$) the probability of the machine concluding that all three people are not lying, and so the complement of that (1-said event) would be the probability of the machine giving positive results for all three?

Any insight would be nice! I am trying to build intuition around these problems.

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Your logic is incorrect. Let us see why. For example, your logic would tell you : The opposite of "it did not rain this week" is "it rained on all days this week". That is not true.

The opposite of "it did not rain this week" is "it rained on atleast one day this week".

Dragging that to your context, the opposite of "all three men were found not lying by the lie detector", is not "all three men were found to be lying by the lie detector", it is "at least one of the three men were found to be lying by the lie detector". This will include the cases : "one man was caught despite not lying", "two men were caught despite not lying", and (but not only!)the case "all three men were caught despite not lying".

With that in mind, let us approach the question.

Reading the question carefully, the probability that "a person telling the truth is caught by the lie detector" is $0.05$.

Hence, the probability that "a person telling the truth is not caught by the lie detector" is $0.95$.

Hence, the probability of "three people all telling the truth not getting caught by the lie detector" is $0.95^3$.

Hence, if the above doesn't happen, then at least one person is caught despite telling the truth. The answer thus is $1 - (0.95^3)$.

If you have not understood, do ask.

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