Question

GrumpyCorp drug tests all of the recent college graduates it hires each year. The drug test currently used correctly determines drug users 96% of the time(a Positive test) and correctly determines non-users 90% of the time(a Negative test). A recent study concluded that 36% of college students use drugs. A potential employee has been tested and the result was Negative for drug use.

a) Construct ALL necessary probabilities using proper notation(Example: P(D) for a "drug user"). (Hint: there should be 6 total)
b) Find the Probability of a Negative test, by showing use of the above Probabilities first, and then followed by the proper calculation.
c) Use Bayes' Theorem to find the probability that a person who tests Positive actually was not a Drug User. Set up using Conditional Probability Notation and then substitute in numeric values.

Answer 1

Drug : Drug user

T : Test positive

a).
`P(D) =0.36`

`$P\left((T)/(D) \right) = 0.96$`

`$P\left((T^c)/(D^c) \right) = 0.90$`

b).
`$P(T^c)= P\left((T^c)/(D^c)\right) * P(D^c)+ P\left((T^c)/(D)\right) * P(D)$`

`$=0.9 * (1-0.36) + (1-0.96) * 0.36$`

= 0.5904

c).
`$P\left((D^c)/(T)\right) = (P\left((T)/(D^c)\right). P(D^c))/(P\left((T)/(D^c)\right). P(D^c) + P\left((T)/(D)\right). P(D))$`

`$=((1-0.90) * (1-0.36))/((1-0.90) * (1-0.36)+(0.96 * 0.36))$`

`$=0.15625$`