Probabilistic Thinking Math Example 1

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Example 1

medium
A medical test is 95% accurate. You test positive. Should you conclude you have the disease? Calculate the probability you actually have it if the disease prevalence is 1%.

Solution

  1. 1
    Given: P(D)=0.01P(D)=0.01, P(+∣D)=0.95P(+|D)=0.95 (sensitivity), P(+∣no D)=0.05P(+|\text{no }D)=0.05 (false positive rate)
  2. 2
    P(+)=P(+∣D)P(D)+P(+∣noD)P(noD)=0.95(0.01)+0.05(0.99)=0.0095+0.0495=0.059P(+) = P(+|D)P(D) + P(+|\text{no}D)P(\text{no}D) = 0.95(0.01) + 0.05(0.99) = 0.0095 + 0.0495 = 0.059
  3. 3
    P(D∣+)=P(+∣D)P(D)P(+)=0.00950.059≈0.161P(D|+) = \frac{P(+|D)P(D)}{P(+)} = \frac{0.0095}{0.059} \approx 0.161
  4. 4
    Despite 95% accurate test, only 16.1% probability of actually having the disease

Answer

P(disease | positive test) ≈ 16.1%. A positive result does not mean you have the disease.
This is Bayes' theorem in action. Low base rate (1% prevalence) means most positive tests are false positives. Probabilistic thinking requires considering base rates, not just test accuracy. This counter-intuitive result is critical for medical decision-making.

About Probabilistic Thinking

Probabilistic thinking is the habit of reasoning about uncertain outcomes in terms of likelihood, expected value, and distributions rather than certainties.

Learn more about Probabilistic Thinking →

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