Mathematics, 20.03.2020 07:47 Jacobstoltzfus

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?

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Given that the person fails the test, the probability that person was telling the truth would be approximately 0.74 (or equivalently, $74\,\%$ .)

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.

$\left\lbrace\begin{aligned}& \text{Truth} && (0.93) \\ & \text{Lie} && (1 - 0.93)\end{aligned}\right.$ .

Assume that the person told the truth, there's 0.14 probability that the person might still fail the test.

$\left\lbrace\begin{aligned}& \text{Truth} && (0.93) \left\lbrace\begin{aligned}& \text{Pass} && (1 - 0.14) \\& \text{Fail} && (0.14)\end{aligned} \right.\\ & \text{Lie} && (1 - 0.93)\end{aligned}\right.$ .

Similarly, assume that the person told the lie, there's a 0.65 probability that the person would fail the test.

$\left\lbrace\begin{aligned}& \text{Truth}\quad (0.93) && \left\lbrace \begin{aligned}& \text{Pass} && (1 - 0.14) \\& \text{Fail} && (0.14)\end{aligned} \right.\\ & \text{Lie} \quad (1 - 0.93) && \left\lbrace\begin{aligned}& \text{Pass} && (1 - 0.65) \\& \text{Fail} && (0.65)\end{aligned} \right.\end{aligned}\right.$ .

Calculate the following:

$P(\text{Truth} \cap \text{Pass})$ . $P(\text{Truth} \cap \text{Fail})$ . $P(\text{Lie} \cap \text{Pass})$ . $P(\text{Lie} \cap \text{Fail})$ .

For example, here's how find $P(\text{Truth} \cap \text{Pass})$ 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 $P(\text{Truth} \cap \text{Pass})$ , the person told the truth. Hence, choose the "Truth" branch. The probability of that branch is 0.93 . Continue to the second branch Pass-Fail branch. In

There are two probabilities along that path: 0.93 and (1 - 0.14) . $P(\text{Truth} \cap \text{Pass})$ is the same as the probability of taking that path. Therefore, $P(\text{Truth} \cap \text{Pass}) = 0.93 \times (1 - 0.14) = 0.7998$ .

Do the same for the other branches.

$P(\text{Truth} \cap \text{Pass}) = 0.93 \times (1 - 0.14) = 0.7998$ . $P(\text{Truth} \cap \text{Fail}) = 0.93 \times 0.14 = 0.1302$ . $P(\text{Lie} \cap \text{Pass}) = (1 - 0.93) \times (1 - 0.65) = 0.0245$ . $P(\text{Lie} \cap \text{Fail}) = (1 - 0.93) \times 0.65 = 0.0455$ ..

The problem is asking for the conditional probability that the person fails the test given that person told the truth. That's written as $P(\text{Truth}\;| \; \text{Fail})$ (the vertical bar stands for "given.") That's a conditional probability. The formula for the conditional probability of A-given-B is:

$\displaystyle P(A\; | \; B) = \frac{P(A \cap B)}{P(B)}$ .

In this case,

$\begin{aligned} & P(\text{Truth}\;| \; \text{Fail}) = \frac{P(\text{Truth}\cap \text{Fail})}{P(\text{Fail})}\end{aligned}$ .

The value of $P(\text{Truth}\cap \text{Fail})$ has already been found. It would be necessary to find $P(\text{Fail})$ .

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

$\begin{aligned}& P(\text{Fail}) \\ &= P(\text{Truth} \cap \text{Fail}) + P(\text{Lie} \cap \text{Fail}) \\ &= 0.1302 + 0.0455 = 0.1757 \end{aligned}$ .

Apply the formula for conditional probabilities:

$\begin{aligned} & P(\text{Truth}\;| \; \text{Fail}) \\ &= \frac{P(\text{Truth}\cap \text{Fail})}{P(\text{Fail})} \\ &= \frac{0.7998}{0.1757} \approx 0.74\end{aligned}$ .

Answer from: Quest

option a is correct.

$\frac{\text{original height}}{\text{original base}}=\frac{\text{reduction height}}{\text{reduction base}}$

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have

$\frac{\text{original height}}{\text{original base}}=\frac{\text{reduction height}}{\text{reduction base}}$

Which is the scale factor proportion for the reduction shown? answers are on picture.

Answer from: Quest

we have three similar triangles, because each has a right angle and shares an angle. let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

$(5 + x)/6 = 6/x$

$x(x+5) = 36$

$x^2 +5x - 36 = 0$

$(x+9)(x-4) = 0$

we reject the negative root and conclude x=4

answer: 4

If bc = 6 and ad = 5, find dc. a) 4 b) 4.5 c) 7.2

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In a lie detector test, 65% of lies are identified as lies and that 14% of true statements are also...