Question

A study by a federal agency concludes that polygraph tests given to truthful persons have probability 0.2 of suggesting that the person is deceptive. A firm asks 12 job applicants about thefts from previous employers, using a polygraph to assess their truthfulness. Suppose that all 12 answer truthfully. Let X be the number of applicants who are classified deceptive. a) Describe the probability mass function of X. b) What is the probability that the polygraph says at least 1 is deceptive? c) What is the mean number among 12 truthful persons who will be classified as deceptive? What is the standard deviation of this number? d) What is the probability that the number classified deceptive is less than the mean?

Answer 1

b) 0.9313

c) mean = 2.4

standard deviation= 1.3856

d) 0.5583

Explanation:

Given:

p = 0.2

n = 12

a) X= number of applicants classified as deceptive.

Probability mass function of X will be:

`P(X=x) = \left(\begin{array}{c}12\\x\end{array}\left) (0.2)^x(1-0.2)^1^2^-^x,x=0, 1, 2, .....,12`

b) Probability that the polygraph says at least 1 is deceptive:

`P(X≥1) = 1 -P(X=0) = 1 -\left(\begin{array}{c}12\\0\end{array}\left) (0.2)^0(1-0.2)^1^2^-^0`

= 1 - 0.0687

= 0.9313

c) The mean number among 12 truthful persons who will be classified as deceptive:

E(X) = n•p

= 12 * 0.2

= 24

Standard deviation:

`s.d = √(12*0.2*(1-0.2))`

= 1.3856

d) Probability that the number classified deceptive is less than the mean:

`P(X<2.4) = P(X≤2) = E^2_x_=_0 \left(\begin{array}{c}12\\0\end{array}\left) (0.2)^0(1-0.2)^1^2^-^0`

= 0.5583