Question

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about \(70 \%\) of cases of neuroblastoma. \({ }^{22}\) It has been proposed that this test be used for large-scale screening of children. Assume that 300,000 children are to be tested, of whom 8 have the disease. We are interested in whether or not the test detects the disease in the 8 children who have the disease. Find the probability that (a) all eight cases will be detected. (b) only one case will be missed. (c) two or more cases will be missed. [Hint: Use parts (a) and (b) to answer part (c).]

Short answer

The probability that (a) all eight cases will be detected is \(0.7^8\), (b) only one case will be missed is \(8 \times 0.7^7 \times 0.3\), and (c) two or more cases will be missed is \(1 - (0.7^8 + 8 \times 0.7^7 \times 0.3)\).

Step-by-step solution

Understand the given probabilities and numbers

First, note the probability of a positive diagnosis with the VMA test, which is 70% or 0.7. Secondly, be aware of the total number of children to be tested, which is 300,000, and the number of children who actually have the disease, which is 8.

Calculate the probability that all eight cases will be detected

Since the probability of detecting the disease in one child is 0.7 and the events are independent, the probability that all eight cases will be detected is the product of the probabilities of each case being detected: \(0.7^8\).

Calculate the probability that exactly one case will be missed

The probability of missing one case is the complement of the detection probability, which is \(1 - 0.7 = 0.3\). Therefore, the probability of detecting 7 cases and missing 1 is \(0.7^7 \times 0.3\), multiplied by 8, since the missed case can be any one of the eight cases.

Calculate the probability that two or more cases will be missed using parts (a) and (b)

To find the probability that two or more cases will be missed, subtract the sum of the probabilities calculated in parts (a) and (b) from 1: \(1 - (0.7^8 + 8 \times 0.7^7 \times 0.3)\).

Key concepts

VMA Test Probability

The VMA test probability is a critical concept when discussing medical diagnostics and its effectiveness in detecting diseases such as neuroblastoma. In our scenario, the test has a 70% chance, or a probability of 0.7, to give a positive diagnosis when neuroblastoma is present. When looking at the probability that the VMA test will detect all eight cases, we use the notion of independent events.

For independent events, which in this context means the outcome of one test does not affect the others, the probability of all events occurring is the product of their individual probabilities. Hence, the likelihood that all eight children with neuroblastoma are detected by the VMA test is calculated by the product of eight individual probabilities, each being 0.7, which is represented mathematically as \(0.7^8\).

Understanding this probability helps health professionals estimate the efficacy of using the VMA test in large-scale screenings and can influence decisions regarding healthcare strategies.

Independent Events in Probability

When discussing independent events in probability, it's key to know that the outcome of one event does not affect another. This is fundamental in calculating the probability of multiple events happening sequentially or concurrently. In the case of our VMA test, we assume that the result of one child's test does not influence the outcome of another's test.

This concept allows us to multiply the probabilities of each event to find the combined probability. For instance, the probability of two independent events both occurring is the product of their probabilities: if Event A has a probability of \(p\) and Event B has a probability of \(q\), then the probability of both A and B occurring is \(p \times q\).

In the exercise, this principle determines the probability that all eight children with neuroblastoma are detected (\(0.7^8\)) as well as the probability of seven detections and one miss (\(8 \times 0.7^7 \times 0.3\)), illustrating the pivotal nature of independent events in probability calculations.

Complement Probability

Complement probability is a foundational idea in probability theory, referring to the likelihood of an event not occurring. It's calculated as one minus the probability that the event does occur. This concept is particularly helpful when it's easier to calculate the chances of an event not happening than it happening.

In our VMA test scenario, this translates to the probability of the test failing to detect neuroblastoma in a child. If the test detects the disease 70% of the time (\(p = 0.7\)), then the probability of it not detecting the disease in a particular case is \(1 - p = 0.3\). The application of the complement rule is seen in calculating the probability of missing exactly one case: we first find the probability of detection, and then use the complement to find the probability of not detecting the disease.

Furthermore, the computation for two or more misses involves understanding that this scenario is the complement of both all cases being detected and one case being missed, a strategy that simplifies the process. Thus, the complement principle is not only intuitive but also a powerful tool in probability calculations.