Bayes' Theorem Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Drug testing: P(user)=0.05P(\text{user})=0.05. Test sensitivity P(+∣user)=0.99P(+|\text{user})=0.99. Specificity P(−∣non-user)=0.95P(-|\text{non-user})=0.95 (so P(+∣non-user)=0.05P(+|\text{non-user})=0.05). Find P(user∣+)P(\text{user}|+).

Solution

  1. 1
    P(non-user)=0.95P(\text{non-user}) = 0.95
  2. 2
    P(+)=P(+∣user)P(user)+P(+∣non-user)P(non-user)=0.99(0.05)+0.05(0.95)=0.0495+0.0475=0.097P(+) = P(+|\text{user})P(\text{user}) + P(+|\text{non-user})P(\text{non-user}) = 0.99(0.05) + 0.05(0.95) = 0.0495 + 0.0475 = 0.097
  3. 3
    P(user∣+)=0.99×0.050.097=0.04950.097≈0.510P(\text{user}|+) = \frac{0.99 \times 0.05}{0.097} = \frac{0.0495}{0.097} \approx 0.510
  4. 4
    Despite 99% sensitivity and 95% specificity, only 51% of positives are true users!

Answer

P(user∣+)≈0.51P(\text{user}|+) \approx 0.51. Even with a highly accurate test, only 51% of positive tests are true users.
The low base rate (5% users) means most positives are false positives, even with a very accurate test. This is the base rate fallacy — ignoring prior probabilities when interpreting test results. Bayes' theorem correctly accounts for prevalence, test sensitivity, and specificity together.

About Bayes' Theorem

Bayes' theorem gives the posterior probability of a hypothesis given evidence: P(H∣E)=P(E∣H)⋅P(H)P(E)P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}.

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More Bayes' Theorem Examples

Example 1 medium

Email spam filter: [formula]. The word 'free' appears in 80% of spam emails and 10% of legitimate em

Example 3 easy

Write out Bayes' theorem and explain each component: [formula].

Example 4 hard

A coin is either fair ([formula], probability 0.7) or biased ([formula], probability 0.3). You flip

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