Question

Composite Sampling for Diabetes Currently, the rate for new cases of diabetes in a year is 3.4 per 1000 (based on data from the Centers for Disease Control and Prevention). When testing for the presence of diabetes, the Portland Diagnostics Laboratory saves money by combining blood samples for tests. The combined sample tests positive if at least one person has diabetes. If the combined sample tests positive, then the individual blood tests are performed.

In a test for diabetes, blood samples from 10 randomly selected subjects are combined. Find the probability that the combined sample tests positive with at least 1 of the 10 people having diabetes. Is it likely that such combined samples test positive?

Short answer

The probability that at least 1 or 10 people test positive for diabetes is 0.0335.

No, it is not likely that combined samples will test positive.

Step-by-step solution

Given Information

The rate at which new cases of diabetes appear in a year is 3.4 per 1000.

The combined sample would test positive if at least oneperson in 10 has diabetes.

The number of subjects who are tested is 10.

Define the probability of an event

Mathematically, the probability of an event A is computed as:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

Let E be the event that a specific sample is positive for diabetes.

The probability that a specific sample is positive is given as follows:

$$\begin{gathered} P\left(E\right)=\frac{3.4}{1000} \\ =0.0034 \end{gathered}$$

Thus, the probability that any sample tests positive is 0.0034.

Step 3:The event of “at least one”

At least one occurrence of an event implies that one or more incidences of the event occur. It is complementary to the event that none of the incidences occurs for an event.

The probability that the event occurs at least once is computed as:

$P\left(\text{at}\text{least}\text{one}\right)=1-P\left(\text{none}\text{occurs}\right)$

Define complementary events

The complementary event for event A is the non-occurrence of the event. The probability of the complement of the event is defined as:

$P\left(\overline{A}\right)=1-P\left(A\right)$

The complement of event E is that the randomly selected sample is not positive.

Thus, the probability is:

$$\begin{gathered} P\left(\overline{E}\right)=1-P\left(E\right) \\ =1-0.0034 \\ =0.9966 \end{gathered}$$

Thus, the probability that a random sample does not test positive is 0.9966.

Define multiplication rule

In the multiplication rule, the probability of co-occurrence for several events is defined as the product of their individual probabilities.

The probability that none of the 10 samples tests positive is computed as:

$$\begin{gathered} P\left(\overline{E}\right)\times P\left(\overline{E}\right)\times ...\times P\left(\overline{E}\right)={\left(P\left(\overline{E}\right)\right)}^{10} \\ ={\left(0.9966\right)}^{10} \\ =0.9665 \end{gathered}$$

Thus, the probability that none of the samples tests positive is 0.9665.

Compute the probability that the combined sample tests positive

The probability that at least one sample is positive is computed as:

$$\begin{gathered} P\left(\text{at}\text{least}\text{one}\text{tests}\text{positive}\right)=1-P\left(\text{none}\text{tests}\text{positive}\right) \\ =1-0.9665 \\ =0.0335 \end{gathered}$$

Thus, the probability that at least one sample is positive in the combined sample is 0.0335.

Interpret the result

The probability that the combined sample tests positive is 0.0335. The value is lesser than 0.05, which implies that the likelihood of getting positive combined samples is very low.