Question

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 positiveif 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.

Anemia Based on data from Bloodjournal.org, 10% of women 65 years of age and older have anemia, which is a deficiency of red blood cells. In tests for anemia, blood samples from 8 women 65 and older are combined. What is the probability that the combined sample tests positive for anemia? Is it likely for such a combined sample to test positive?

Short answer

The probability that the combined sample tests positive for anemiais equal to 0.570.

It is likely that such a combined sample will test positive for anemia as the value is quite high.

Step-by-step solution

Given information

It is given that a combined blood sample of 8women will test positive for anemia if at least one of the eight samples tests positive.

Ten percent of women aged 65 and above are positive for anemia.

Required probability 

Let X denote 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:

$$\begin{gathered} p=10\% \\ =\frac{10}{100} \\ =0.1 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.10 \\ =0.90 \end{gathered}$$

The number of trials (n) is equal to 8.

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:

$$\begin{gathered} P\left(X⩾1\right)=1-P\left(X<1\right) \\ =1-P\left(X=0\right) \\ =1-{}^8{C}_0{\left(0.10\right)}^0{\left(0.90\right)}^{8-0} \\ =1-0.430467 \\ =0.570 \end{gathered}$$

Thus, the probability that the combined sample tests positive for anemia is equal to 0.570.

Since the value is quite high (greater than 50%), it is likely for such a combined sample to test positive for anemia.