Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

If a certain disease is present, then a blood test will reveal it \(95 \%\) of the time. But the test will also indicate the presence of the disease \(2 \%\) of the time when in fact the person tested is free of that disease; that is, the test gives a false positive \(2 \%\) of the time. If \(0.3 \%\) of the general population actually has the disease, what is the probability that a person chosen at random from the population has the disease given that he or she tested positive?

Step-by-Step Solution

Answer

The probability that a person chosen at random from the population has the disease given that he or she tested positive is approximately 9.85%.

Step 1: Understand the problem and notations

Let's assign some notations to the given information: - A: event that a person has the disease. - B: event that a person tests positive for the disease. We want to find the conditional probability P(A|B) - that is, the probability that a person has the disease given that he or she tested positive.

Step 2: Apply Bayes' Theorem

Bayes' theorem relates the conditional probabilities P(A|B) and P(B|A). It is given by the following formula: \(P(A|B) = \frac{P(B|A) * P(A)}{P(B)}\) Here, we have P(B|A) which is 0.95, the probability of a person testing positive given that they have the disease, and P(A) which is 0.003, the probability of a person having the disease.

Step 3: Find P(B)

To find P(B), the probability that a person tests positive regardless of whether they have the disease or not, we can use the Law of Total Probability: \(P(B) = P(B|A) * P(A) + P(B|\overline{A}) * P(\overline{A})\) Here, \(\overline{A}\) represents the event that a person does not have the disease. We are given P(B|\(\overline{A}\)) = 0.02, which is probability of a false positive.

Step 4: Calculate P(\(\overline{A}\))

Since A is the event that a person has the disease and P(A)=0.003, the probability of a person not having the disease is: \(P(\overline{A}) = 1 - P(A) = 1 - 0.003 = 0.997\)

Step 5: Calculate P(B)

Using the values obtained, we can now calculate P(B): \(P(B) = P(B|A) * P(A) + P(B|\overline{A}) * P(\overline{A}) = 0.95 * 0.003 + 0.02 * 0.997 = 0.02885 \)

Step 6: Calculate P(A|B)

Now that we have P(B), we can use Bayes' theorem to find P(A|B): \(P(A|B) = \frac{P(B|A) * P(A)}{P(B)} = \frac{0.95 * 0.003}{0.02885} = 0.09847 \) Therefore, the probability that a person chosen at random from the population has the disease given that he or she tested positive is approximately 9.85%.

Key Concepts

Conditional Probability

Understanding conditional probability is crucial for solving problems that involve the likelihood of an event occurring given that another event has already occurred. Technically, the conditional probability of an event can be seen as updating our beliefs about the likelihood of one event based on the occurrence of another.

For instance, in our exercise, the event of interest is a person having the disease given that the blood test is positive. This is expressed as the conditional probability P(A|B), where A is the event of having the disease, and B is the event of testing positive. The challenge here is to avoid assuming that a positive test singularly confirms the disease without considering false positives and the overall prevalence of the disease in the population.

False Positive

A 'false positive' refers to a situation wherein a test incorrectly indicates the presence of a condition, such as a disease when it's actually not present. In medical testing, a false positive can cause undue stress and can lead to unnecessary treatments. The probability of a false positive is represented as P(B|\(\bar{A}\)) in our exercise, where \(\bar{A}\) signifies not having the disease.

Medical tests are rarely perfect, and thus accounting for false positives is vital in evaluating the reliability of a test's results. In our problem, even though a small percentage of the healthy population might receive positive test results, incorporating this figure into Bayes' Theorem provides a more accurate picture of a test's predictive value.

Total Probability Law

The law of total probability connects marginal probabilities to conditional probabilities. It's a fundamental rule in probability that allows us to break down complex probabilities into simpler components. In mathematical terms, it can be expressed as:

\[P(B) = P(B|A) \cdot P(A) + P(B|\bar{A}) \cdot P(\bar{A})\]
Where:

• P(B|A) is the probability of B given A.
• P(A) is the probability of A.
• P(B|\bar{A}) is the probability of B given not A.
• P(\bar{A}) is the probability of not A.

This approach is valuable when considering all possible scenarios that lead to a specific outcome — in our case, all the ways someone can test positive, with and without the disease. It's an indispensable component when using Bayes' Theorem as it informs the denominator necessary for the final conditional probability calculation.