Answer:
The probability that an uses drugs given that the result was negative is 0.0224.
Step-by-step explanation:
Let's denote the events as follows:
D = an intern uses drugs.
X = the test result was positive.
The information provided is:
[tex]P(X|D)=0.80\\P(X|D^{c})=0.03\\P(D)=0.10[/tex]
Compute the probability that the test result of an intern is positive as follows:
[tex]P(X)=P(X|D)P(D)+P(X|D^{c})P(D^{c})\\=P(X|D)P(D)+P(X|D^{c})[1-P(D^{c})]\\=(0.80\times0.10)+(0.03\times(1-0.10))\\=0.107[/tex]
The probability that the test result of an intern is positive is P (X) = 0.107.
Compute the probability that the result is negative given that he/she uses drugs as follows:
[tex]P(X^{c}|D)=1-P(X|D)=1-0.80=0.20[/tex]
Compute the probability that an uses drugs given that the result was negative as follows:
[tex]P(D|X^{c})=\frac{P(X^{c}|D)P(D)}{P(X^{c})}=\frac{P(X^{c}|D)P(D)}{1-P(X)} =\frac{0.20\times0.10}{1-0.107} =0.0224[/tex]
Thus, the probability that an uses drugs given that the result was negative is 0.0224.
