# Biostatistics | Probability homework help

A study was conducted to assess the sensitivity and specicity of four dif-

ferent human immunodeciency virus (HIV) serology tests (Koblavi-Deme

et al. 2001). The Determine test was among the four, it was developed by

Abbott Laboratories (an American provider of health care, medical devices

and pharmaceuticals) and was found to have a true negative rate (the true

negative rate is also called specicity) of 99.4% and a true positive rate (the

true positive rate is also called sensitivity) of 100%. The true negative rate

of a test for a disease is the probability that someone without the disease

tests negative. The true positive rate of a test for a disease is the probabil-

ity that someone with the disease tests positive. HIV may be transmitted

from an expecting parent to their child by transmission during childbirth

or by transmission to the fetus during pregnancy (throughout, assume that

there’s no other way for a newborn to be infected). Treatment by the drugs

zidovudine or nevirapine has been shown to reduce the rate of these sorts

of transmission of HIV by 38% to 50% in the absence of other intervention

(Koblavi-Deme et al. 2001).

a) Suppose that an expecting parent is infected with HIV and they are

treated with zidovudine or nevirapine during pregnancy. Suppose that

after they give birth, a Determine serology test reports a positive test

for HIV. What is the probability that the child does not have HIV?

Round your answer to the nearest 10-th of a percent.

b) UNAIDS (an organization established by the United Nations Economic

and Social Council) estimates the prevalence of HIV in C^ote d’Ivoire

among people aged 15-49 to be 2.6%. If a Determine serology test re-

ported a positive test for HIV in someone selected uniformly at random

among all people in C^ote d’Ivoire aged 15-49, what is the probability

that the person does not have HIV? Round your answer to the nearest

10-th of a percent.

c) In the USA, according to the Centers for Disease Control (a public

health institute within the United States Department of Health and

Human Services), if someone has a positive serology test for HIV they

are not diagnosed as HIV-positive until a second follow-up test also

yields a positive test result. What is the probability that someone is

incorrectly diagnosed as HIV-positive (i.e., if someone is not infected

with HIV, what is the probability that their rst test and also their

second follow-up test are both positive)? Suppose that both tests are

Determine serology tests, and also assume that the test results are

statistically independent. Express your answer in expected number of

events in a million (i.e. something like `a 36 in a million chance’ or

`a one in a million chance’). Also: In one sentence, what is a possible

argument as to why the assumption of independence of the two test

results might be wrong? (Your argument does not have to be sound,

but it must be valid without being tautological).

d) What is the probability that an HIV infected expecting parent trans-

mits HIV to their child either during childbirth or through transmit-

ting HIV to the fetus during pregnancy, given that the parent has not

received treatment with the drugs zidovudine or nevirapine, and in

the absence of other intervention, according to the preamble of this

problem (in concordance with Koblavi-Deme et al. 2001)?