Respuesta :
Answer:
(a) The probability that the tested individual uses this illegal drug given that the result is positive is 0.5213.
(b) The probability of a false positive is 0.10.
(c) The probability of a false negative is 0.02.
Step-by-step explanation:
Let the events be denoted as follows:
D = an individual uses the illegal drug
X = the drug test result is positive.
The information provided is:
[tex]P(X|D) = 0.98\\P(X^{c}|D^{c})=0.90\\P(D)=0.10[/tex]
(a)
Compute the probability that the tested individual uses this illegal drug given that the result is positive as follows:
[tex]P(D|X)=\frac{P(X|D)P(D)}{P(X)}[/tex]
Compute the probability of the test result being positive as follows:
[tex]P(X)=P(X|D)P(D)+P(X|D^{c})P(D^{c})\\=P(X|D)P(D)+[1-P(X^{c}|D^{c})][1-P(D)]\\=(0.98\times0.10)+[(1-0.90)(1-0.10)]\\=0.188[/tex]
The probability that the tested individual uses this illegal drug given that the result is positive is:
[tex]P(D|X)=\frac{P(X|D)P(D)}{P(X)}=\frac{0.98\times0.10}{0.188} =0.5213[/tex]
Thus, the probability that the tested individual uses this illegal drug given that the result is positive is 0.5213.
(b)
Compute the probability of a false positive given that the result was positive as follows:
[tex]P(X|D^{c})=1-P(X^{c}|D^{c})=1-0.90=0.10[/tex]
Thus, the probability of a false positive is 0.10.
(c)
Compute the probability of a false negative as follows:
[tex]P(X^{c}|D)=1-P(X|D)=1-0.98=0.02[/tex]
Thus, the probability of a false negative is 0.02.
