In a lie detector test, 65% of lies are identified as lies and that 14% of true statements are also identified as lies. 93% of the job applicants tell the truth during the polygraph test. What is the probability that a person who fails the test was actually telling the truth?

Given that the person fails the test, the probability that person was telling the truth would be  approximately (or equivalently, .)  

Step-by-step explanation:

Start by drawing a probability tree. What the person said would affect the outcome of the test. As a result, the first split in this tree should be based on whether the person told the truth or not.

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Assume that the person told the truth, there's probability that the person might still fail the test.

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Similarly, assume that the person told the lie, there's a probability that the person would fail the test.

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Calculate the following:

For example, here's how find using the tree. (That's the probability that the person told the truth and passed the test.)

Start from the Truth-Lie split at the left-hand side. In , the person told the truth. Hence, choose the "Truth" branch. The probability of that branch is . Continue to the second branch Pass-Fail branch. In

There are two probabilities along that path: and . is the same as the probability of taking that path. Therefore, .

Do the same for the other branches.

The problem is asking for the conditional probability that the person fails the test given that person told the truth. That's written as (the vertical bar stands for "given.") That's a conditional probability. The formula for the conditional probability of A-given-B is:

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In this case,

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The value of has already been found. It would be necessary to find .

That person can fail the test either after telling a truth or after telling a lie. There's no other way to fail the test. That means

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Apply the formula for conditional probabilities:

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step-by-step explanation:

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step-by-step explanation: