Question

Grumpy Corp 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.

Required:
a. Construct ALL necessary probabilities using proper notation(Example: P(D) for a "drug user").
b. Find the Probability of a Negative test, by showing use of the above Probabilities first, and then followed by the proper calculation.

Answer 1

Hence, the probability of a Negative test is
`P(T^C)=0.5904`.

Explanation:

(a)

Let
`D=` Drug user and
`T=` Test positive

Probability of drug users
`P(D)=36\%=0.36`

`P((T)/(D))=96\%=0.96`

`P((T^c)/(D^c))=90\%=0.90`

(b)

Find the Probability of a Negative test,

`P(T^C)=P((T^C)/(D^C))\cdot P(D^C)+P((T^C)/(D))\cdot P(D)`

`\Rightarrow P(T^C)=0.9\cdot(1-0.36)+(1-0.96)\cdot 0.36`

`\Rightarrow P(T^C)=0.9\cdot 0.64+0.04\cdot 0.36`

`\Rightarrow P(T^C)=0.576+0.0144`

`\Rightarrow P(T^C)=0.5904`