Composite Sampling. Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.

HIV It is estimated that worldwide, 1% of those aged 15–49 are infected with the human immunodeficiency virus (HIV) (based on data from the National Institutes of Health). In tests for HIV, blood samples from 36 people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

It is given that a combined blood sample of 36 people will test positive for HIV if at least one of the 36 samples tests positive.

One percent of the people in the world between the ages 15-49 years are positive for HIV.

Let Xdenote the number of samples that test positive in the combined sample.

Success is defined as getting a positive sample in the combined sample.

The probability of success is computed below:

$$\begin{gathered} p=1\% \\ =\frac{1}{100} \\ =0.01 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.01 \\ =0.99 \end{gathered}$$

The number of trials (n) is equal to 36.

The binomial probability formula is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

By using the binomial probability formula, the probability of at least one positive sample is computed below:

$$\begin{gathered} P\left(X⩾1\right)=1-P\left(X<1\right) \\ =1-P\left(X=0\right) \\ =1-{}^{36}{C}_0{\left(0.01\right)}^0{\left(0.99\right)}^{36-0} \\ =1-0.696413 \\ =0.304 \end{gathered}$$

Thus, the probability that the combined sample tests positive for HIV is equal to 0.304.

Since the value is not low, it is not unlikely for such a combined sample to test positive.