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

Answer:

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

Step-by-step explanation:

(a)

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

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

                                    $P(\frac{T}{D})=96\%=0.96$

                                   $P(\frac{T^c}{D^c})=90\%=0.90$

(b)

Find the Probability of a Negative test,

     $P(T^C)=P(\frac{T^C}{D^C})\cdot P(D^C)+P(\frac{T^C}{D})\cdot P(D)$

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

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

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

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