Answer:
(a) The probability tree is shown below.
(b) The probability that a person who does not use heroin in this population tests positive is 0.10.
(c) The probability that a randomly chosen person from this population is a heroin user and tests positive is 0.0279.
(d) The probability that a randomly chosen person from this population tests positive is 0.1249.
(e) The probability that a person is heroin user given that he/she was tested positive is 0.2234.
Step-by-step explanation:
Denote the events as follows:
<em>X</em> = a person is a heroin user
<em>Y</em> = the test is correct.
Given:
P (X) = 0.03
P (Y|X) = 0.93
P (Y|X') = 0.99
(a)
The probability tree is shown below.
(b)
Compute the probability that a person who does not use heroin in this population tests positive as follows:
The event is denoted as (Y' | X').
Consider the tree diagram.
The value of P (Y' | X') is 0.10.
Thus, the probability that a person who does not use heroin in this population tests positive is 0.10.
(c)
Compute the probability that a randomly chosen person from this population is a heroin user and tests positive as follows:
![P(X\cap Y)=P(Y|X)P(X)=0.93\times0.03=0.0279](https://tex.z-dn.net/?f=P%28X%5Ccap%20Y%29%3DP%28Y%7CX%29P%28X%29%3D0.93%5Ctimes0.03%3D0.0279)
Thus, the probability that a randomly chosen person from this population is a heroin user and tests positive is 0.0279.
(d)
Compute the probability that a randomly chosen person from this population tests positive as follows:
P (Positive) = P (Y|X)P(X) + P (Y'|X')P(X')
![=(0.93\times0.03)+(0.10\times0.97)\\=0.1249](https://tex.z-dn.net/?f=%3D%280.93%5Ctimes0.03%29%2B%280.10%5Ctimes0.97%29%5C%5C%3D0.1249)
Thus, the probability that a randomly chosen person from this population tests positive is 0.1249.
(e)
Compute the probability that a person is heroin user given that he/she was tested positive as follows:
![P(X|positive)=\frac{P(Y|X)P(X)}{P(positive)} =\frac{0.93\times0.03}{0.1249}= 0.2234](https://tex.z-dn.net/?f=P%28X%7Cpositive%29%3D%5Cfrac%7BP%28Y%7CX%29P%28X%29%7D%7BP%28positive%29%7D%20%3D%5Cfrac%7B0.93%5Ctimes0.03%7D%7B0.1249%7D%3D%200.2234)
Thus, the probability that a person is heroin user given that he/she was tested positive is 0.2234.