Probability and Clinical Reasoning

Why Experienced Clinicians Order Fewer Tests — and Get More Answers

Learning Objectives

  1. Recognise that history → examination → investigation IS Bayesian updating
  2. Apply the complement, addition, and multiplication rules
  3. Distinguish independent from dependent events
  4. Understand why P(Disease|Test+) \(\neq\) P(Test+|Disease)
  5. Convert between probability and odds
  6. Apply Bayes’ theorem using natural frequencies
  7. Trace a diagnostic cascade as sequential updates
  8. Identify the base rate, prosecutor’s, and gambler’s fallacies

The CBC Mystery in Pathology

Monthly lab audit finding:

CBCs ordered by senior faculty → 40% abnormality rate

CBCs ordered by PG residents → 12% abnormality rate

Are residents ordering unnecessary tests? Are faculty missing patients?

. . .

Neither. This is Bayes’ theorem at work in the hospital corridors.

Faculty order CBCs when clinical assessment has already raised suspicion — high pre-test probability.

Residents cast a wider net — low pre-test probability.

Same test. Same lab. Different yield.

You Already Think Like Bayes

Faculty order tests at steps 3-4 (high prior). Residents order at step 1 (low prior). That’s the CBC mystery solved.

Three Basic Probability Rules

Rule 1: Complement

\[P(\text{not } A) = 1 - P(A)\]

Drug causes nausea in 15% → P(no nausea) = 85%

Rule 2: Addition (OR)

Mutually exclusive: \[P(A \text{ or } B) = P(A) + P(B)\]

Non-mutually exclusive: \[P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)\]

Screening camp: DM=12%, HTN=25%, Both=6% \[P(\text{DM or HTN}) = 12 + 25 - 6 = 31\%\]

Rule 3: Multiplication (AND)

Independent: \[P(A \text{ and } B) = P(A) \times P(B)\]

Dependent: \[P(A \text{ and } B) = P(A) \times P(B|A)\]

Blood culture contamination = 3% per bottle. \[P(\text{both contaminated}) = 0.03 \times 0.03 = 0.09\%\]

This is why 2 positive cultures are far more convincing than 1!

Visualising the Addition Rule

Independence vs Dependence

Scenario Type Why?
Two blood cultures from the same bacteraemic patient Dependent Both bottles grow the same organism
Blood cultures from two unrelated patients Independent One doesn’t cause the other
Side effects in Patient A and Patient B Independent Different biology
Two joints in same RA patient Dependent Systemic disease

Clinical Caution

Two bottles drawn from the same arm are NOT independent — contamination is correlated. Guidelines require separate venepuncture sites.

Conditional Probability: The Diagnostic Trap

\[P(A|B) = \frac{P(A \text{ and } B)}{P(B)}\]

The two most critical conditional probabilities in medicine:

What we know What we want Name Who uses it?
Patient has disease P(Test+ | Disease) Sensitivity Lab scientists
Test is positive P(Disease | Test+) PPV Clinicians

The Most Dangerous Confusion in Medicine

\[P(\text{Disease} | \text{Test+}) \neq P(\text{Test+} | \text{Disease})\]

Sensitivity is about the test. PPV is about the patient.

A test can be 95% sensitive and have a PPV of only 16% if the disease is rare.

The CBC Mystery Explained

Faculty aren’t better at reading CBCs. They’re better at selecting who needs one.

Probability vs Odds

Conversion: \[\text{Odds} = \frac{P}{1-P} \qquad P = \frac{\text{Odds}}{1+\text{Odds}}\]

Probability Odds Clinical Example
1% 1:99 Breast ca. screening
10% 1:9 TB, symptomatic (MP)
20% 1:4 PE, moderate Wells
50% 1:1 No information
80% 4:1 MI with ECG changes
90% 9:1 2x blood culture+

Why Odds Matter

\[\text{Post-test odds} = \text{Pre-test odds} \times \text{Likelihood Ratio}\]

One multiplication. That’s the entire Bayesian update.

Bayes’ Theorem: Two Forms

Probability form (complex):

\[P(D|T^+) = \frac{P(T^+|D) \cdot P(D)}{P(T^+|D) \cdot P(D) + P(T^+|\bar{D}) \cdot P(\bar{D})}\]

Components:

  • \(P(D)\) = pre-test probability
  • \(P(T^+|D)\) = sensitivity
  • \(P(T^+|\bar{D})\) = 1 - specificity

Odds form (simple!):

\[\text{Post-test odds} = \text{Pre-test odds} \times LR\]

Likelihood Ratios:

\[LR^+ = \frac{\text{Sensitivity}}{1-\text{Specificity}}\]

\[LR^- = \frac{1-\text{Sensitivity}}{\text{Specificity}}\]

Natural frequencies make this even easier — just think in 1000 patients.

Dengue NS1 Test: Monsoon vs January

Same test, same lab, same technician. PPV drops from 75% to 7% when prevalence changes.

PPV vs Prevalence: The Whole Picture

The Mammography Dilemma

Gigerenzer showed that presenting natural frequencies raised doctors’ accuracy from ~15% to ~85%.

The SLE Diagnostic Cascade

ANA (Sens 95%, Spec 60%) is for screening. Anti-dsDNA (Sens 60%, Spec 95%) is for confirmation.

Ruling OUT Pulmonary Embolism

Bayesian reasoning saves the patient from unnecessary radiation, contrast dye, and cost.

Three Probability Fallacies

Base rate fallacy is #1: A 95% sensitive COVID test at 2% prevalence gives PPV of only 16%.

Bayesian Update: Worked Example

Patient: 20% pre-test probability of PE. D-dimer negative (LR- = 0.1).

Step Calculation Result
Pre-test odds 0.20 / 0.80 0.25
Post-test odds 0.25 \(\times\) 0.1 0.025
Post-test probability 0.025 / 1.025 2.4%

Clinical Implication

Negative D-dimer drops PE probability from 20% to ~2.4% — safe to rule out without CT angiography. This is Bayes at the bedside.

Key Takeaways

  1. Clinical diagnosis IS Bayesian reasoning — history → examination → investigation is sequential probability updating

  2. Pre-test probability is king — it’s why faculty CBCs yield more abnormals than residents’ CBCs

  3. P(Disease|Test+) \(\neq\) P(Test+|Disease) — sensitivity is NOT PPV

  4. Natural frequencies (“think in 1000 patients”) make Bayes intuitive

  5. Post-test odds = Pre-test odds \(\times\) LR — one formula for bedside reasoning

  6. The base rate fallacy is the most dangerous probability error in clinical medicine

Further Learning

Videos:

Books:

  • Gigerenzer G. Reckoning with Risk. Penguin; 2002.
  • Sackett DL et al. Clinical Epidemiology. 3rd ed. Ch 4.
  • Bland M. An Introduction to Medical Statistics. 4th ed. Ch 20.