Question

In a large population, 3% of the people are heroin users. A new drug test correctly identifies users 93% of the time and correctly identifies nonusers 90% of the time.Answer the following, showingall work for fullcredit.

a) Draw the probability tree for this scenario. Label the outcomes and indicate
b) What is the probability that a person who does not use heroin in this
c) What is the probability that a randomly chosen person from this population
d) What is the probability that a randomly chosen person from this population
e) If a person tests positive for heroin, what is the probability that he/she is a all the probabilities on the tree. population tests positive? is a heroin user and tests positive? tests positive? heroin user?

Answer 1

(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.

Explanation:

Denote the events as follows:

X = a person is a heroin user

Y = 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*0.03=0.0279`

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*0.03)+(0.10*0.97)\\=0.1249`

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)=(P(Y|X)P(X))/(P(positive)) =(0.93*0.03)/(0.1249)= 0.2234`

Thus, the probability that a person is heroin user given that he/she was tested positive is 0.2234.