{"id":1640,"date":"2026-03-05T00:57:01","date_gmt":"2026-03-05T00:57:01","guid":{"rendered":"https:\/\/www.think10x.ai\/?p=1640"},"modified":"2026-03-05T00:57:17","modified_gmt":"2026-03-05T00:57:17","slug":"bayes-theorem-example","status":"publish","type":"post","link":"https:\/\/www.think10x.ai\/blog\/bayes-theorem-example\/","title":{"rendered":"How to Use Bayes Theorem to Calculate Disease Probability After a Positive Test?"},"content":{"rendered":"\t\t
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Question<\/h2>

A medical test is 95%<\/strong> sensitive and 90%<\/strong> specific. Only 2%<\/strong> of the population has the disease. If someone tests positive, what is the probability they are actually sick<\/strong>?<\/span><\/p>

Quick Answer<\/h2>

Using Bayes Theorem<\/strong>, the probability that a person actually has the disease after testing positive is approximately 16.24%<\/strong>. This might seem low, but it makes sense once you factor in how rare the disease is. With only 2%<\/strong> prevalence, most of the positive results<\/strong> come from healthy people<\/strong>, not sick ones. Watch the video below to see exactly how we get there.<\/span><\/p>

Bottom Line<\/h3>

P (Disease | Positive Test) = 16.24%<\/strong><\/p>

Even with a 95%<\/strong> sensitive test, low disease prevalence means only about 1 in 6<\/strong> positive results is a true positive.<\/span><\/p>

How to Solve a Bayes Theorem Problem: Video Walkthrough<\/h2>

We created a step-by-step video walkthrough of this Bayes theorem<\/strong> example. Watch how we apply the Bayes rule formula from scratch, building each piece of the equation before combining them into the final answer.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t

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Full Video Transcript<\/h2>

Below is the complete transcript<\/strong> from the video explanation<\/strong>:<\/span><\/p>

  1. Let’s read the problem together. We’re given sensitivity<\/strong>, specificity<\/strong>, and prevalence<\/strong>, and we want the probability that a person truly has the disease given a positive test. We’ll use Bayes’ theorem<\/strong> and work slowly, step by step.<\/span><\/li>
  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.<\/span><\/li>
  3. Now, we translate the prevalence<\/strong> into a probability<\/strong>. 2%<\/strong> have the disease. Sensitivity is the chance the test is positive when the person truly has the disease (95%<\/strong>). Specificity<\/strong> is the chance the test is negative when the person does not have the disease (90%<\/strong>).<\/span><\/li>
  4. From specificity, we can get the false positive rate<\/strong> using the complement: 1 minus specificity<\/strong>. Similarly, if 2%<\/strong> have the disease, then 98%<\/strong> do not, which is the complement of the prevalence.<\/span><\/li>
  5. Great, now we apply Bayes’ theorem to combine the prior probability<\/strong> with test accuracy<\/strong>. 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<\/strong> is large relative to the true positive part.<\/span><\/li>
  6. Add those to get the total chance of a positive result. Finally, divide the numerator<\/strong> by the denominator<\/strong> to get the posterior probability<\/strong> that a person truly has the disease after a positive test.<\/span><\/li>
  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%<\/strong>. Nice work stepping carefully through Bayes’ theorem.<\/span><\/li><\/ol>

    Step-by-Step Explanation<\/h2>

    Bayes theorem<\/strong> 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.<\/p>

    Step 1: Define the Events<\/h3>

    Before plugging in any numbers, let\u2019s clearly name what we\u2019re 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<\/strong> or negative<\/strong>.<\/p>