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:
<em>D</em> = an individual uses the illegal drug
<em>X</em> = the drug test result is positive.
The information provided is:
![P(X|D) = 0.98\\P(X^{c}|D^{c})=0.90\\P(D)=0.10](https://tex.z-dn.net/?f=P%28X%7CD%29%20%3D%200.98%5C%5CP%28X%5E%7Bc%7D%7CD%5E%7Bc%7D%29%3D0.90%5C%5CP%28D%29%3D0.10)
(a)
Compute the probability that the tested individual uses this illegal drug given that the result is positive as follows:
![P(D|X)=\frac{P(X|D)P(D)}{P(X)}](https://tex.z-dn.net/?f=P%28D%7CX%29%3D%5Cfrac%7BP%28X%7CD%29P%28D%29%7D%7BP%28X%29%7D)
Compute the probability of the test result being positive as follows:
![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](https://tex.z-dn.net/?f=P%28X%29%3DP%28X%7CD%29P%28D%29%2BP%28X%7CD%5E%7Bc%7D%29P%28D%5E%7Bc%7D%29%5C%5C%3DP%28X%7CD%29P%28D%29%2B%5B1-P%28X%5E%7Bc%7D%7CD%5E%7Bc%7D%29%5D%5B1-P%28D%29%5D%5C%5C%3D%280.98%5Ctimes0.10%29%2B%5B%281-0.90%29%281-0.10%29%5D%5C%5C%3D0.188)
The probability that the tested individual uses this illegal drug given that the result is positive is:
![P(D|X)=\frac{P(X|D)P(D)}{P(X)}=\frac{0.98\times0.10}{0.188} =0.5213](https://tex.z-dn.net/?f=P%28D%7CX%29%3D%5Cfrac%7BP%28X%7CD%29P%28D%29%7D%7BP%28X%29%7D%3D%5Cfrac%7B0.98%5Ctimes0.10%7D%7B0.188%7D%20%3D0.5213)
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:
![P(X|D^{c})=1-P(X^{c}|D^{c})=1-0.90=0.10](https://tex.z-dn.net/?f=P%28X%7CD%5E%7Bc%7D%29%3D1-P%28X%5E%7Bc%7D%7CD%5E%7Bc%7D%29%3D1-0.90%3D0.10)
Thus, the probability of a false positive is 0.10.
(c)
Compute the probability of a false negative as follows:
![P(X^{c}|D)=1-P(X|D)=1-0.98=0.02](https://tex.z-dn.net/?f=P%28X%5E%7Bc%7D%7CD%29%3D1-P%28X%7CD%29%3D1-0.98%3D0.02)
Thus, the probability of a false negative is 0.02.