How to Use Bayes Theorem to Calculate Disease Probability After a Positive Test?

Full Video Transcript

Below is the complete transcript from the video explanation:

1. Let’s read the problem together. We’re given sensitivity, specificity, and prevalence, and we want the probability that a person truly has the disease given a positive test. We’ll use Bayes’ theorem and work slowly, step by step.
2. This low prevalence will heavily shape the final probability. This is the target probability we need to compute. First, let’s name the events so we can write everything clearly. This keeps our reasoning tidy.
3. Now, we translate the prevalence into a probability. 2% have the disease. Sensitivity is the chance the test is positive when the person truly has the disease (95%). Specificity is the chance the test is negative when the person does not have the disease (90%).
4. From specificity, we can get the false positive rate using the complement: 1 minus specificity. Similarly, if 2% have the disease, then 98% do not, which is the complement of the prevalence.
5. Great, now we apply Bayes’ theorem to combine the prior probability with test accuracy. Let’s compute each piece. First, the numerator: Positive given disease times the prevalence. Next, the false positive contribution in the denominator. This false positive contribution is large relative to the true positive part.
6. Add those to get the total chance of a positive result. Finally, divide the numerator by the denominator to get the posterior probability that a person truly has the disease after a positive test.
7. So even with a sensitive and fairly specific test, because the disease is rare, the chance of truly having it after a positive is about 16%. Nice work stepping carefully through Bayes’ theorem.

Step-by-Step Explanation

Bayes theorem is really asking one simple question about how many people who test positive actually have the disease. That framing makes the math much more intuitive. Here is how we use Bayes theorem to solve this problem step by step.

Step 1: Define the Events

Before plugging in any numbers, let’s clearly name what we’re working with. Think of it like setting up a confusion matrix with four possible outcomes depending on whether the person has the disease and whether the test fires positive or negative.

• D = person has the disease
• +  = test result is positive
• We want P(D | +), the probability of having the disease given a positive test

Step 2: Extract the Given Probabilities

This is where prevalence becomes critical. Only 2% of the population has this disease, which means 98% of people who walk in for a test are healthy. That imbalance is what makes the result so surprising.

• P(Disease) = 0.02 <- 2% prevalence (only 2 in 100 people are sick)
• P(No Disease) = 0.98 <- 98% are healthy
• P(+ | Disease) = 0.95 <- 95% sensitivity; test catches most true cases
• P(+ | No Disease) = 0.10 <- 10% false positive rate (1 – 90% specificity)

Notice that the 10% false positive rate applied to 98 healthy people generates far more false alarms than the true positives from just 2 sick people.

Step 3: Apply the Bayes Rule Formula

This is where the Bayes rule formula comes together. It combines the prior probability of disease with how likely the test is to be positive, producing the Bayes theorem probability for our specific scenario. Think of the denominator as asking how many people out of 100 will test positive, sick or not?

• The formula is P(D | +)  =  P(+ | D) x P(D)  /  P(+)
• Where the total probability of a positive result P(+) accounts for both true positives and false positives: P(+) = P(+ | D) x P(D)  +  P(+ | No D) x P(No D)

Step 4: Plug in the Numbers

Let’s put real numbers in and see what happens:

• Numerator (true positives): 0.95 x 0.02 = 0.019
• Denominator (all positives): (0.95 x 0.02) + (0.10 x 0.98) = 0.019 + 0.098 = 0.117
• Result: P(Disease | Positive) = 0.019 / 0.117 = 0.1624 = 16.24%

The aha moment is that even though the test is 95% sensitive, 0.098 of the population are healthy people who still trigger a positive. That false positive pool is five times larger than the true positive pool (0.019), which is why only about 1 in 6 positive results is genuine. This is the posterior probability, the updated belief after receiving a positive test result. The low prevalence of 2% is the key driver.

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