Bayes' Theorem: Is Smoking Independent of Lung Cancer?

Probability & Statistics 11th-12th Grade
Problem
Large-scale surveys indicate that 11% of the population smokes. Medical researchers know that the probability a smoker will get lung cancer is 0.34. The probability a person will get lung cancer if they don't smoke is 0.03. What is the probability that a lung cancer patient is a smoker? Are smoking and lung cancer independent events?

What You Will Learn

  • Bayes' theorem application — converting from P(cancer|smoker) to P(smoker|cancer)
  • Law of total probability — calculating overall disease rates from conditional probabilities
  • Independence testing — determining when knowing one event changes the probability of another
  • Medical probability interpretation — understanding what conditional probabilities mean in real health contexts
  • Base rate importance — seeing how population percentages affect individual diagnostic probabilities

Solution: Method 1 — Direct Bayes' Theorem Application

This is a classic "reverse conditional probability" problem. We know disease rates for smokers vs. non-smokers, but we want to know smoking rates among cancer patients. This requires Bayes' theorem.

Step 1 — Define events and organize given information

Let S = person is a smoker, and C = person gets lung cancer.

Given:
P(S) = 0.11 (11% of population smokes)
P(S') = 0.89 (89% don't smoke)
P(C|S) = 0.34 (34% of smokers get lung cancer)
P(C|S') = 0.03 (3% of non-smokers get lung cancer)

Want: P(S|C) and test independence

Step 2 — Calculate the total probability of cancer P(C)

Using the law of total probability, the overall cancer rate is the weighted average of cancer rates in each group:

P(C) = P(C|S) × P(S) + P(C|S') × P(S')
P(C) = 0.34 × 0.11 + 0.03 × 0.89
P(C) = 0.0374 + 0.0267
P(C) = 0.0641

So 6.41% of the total population will develop lung cancer.

Step 3 — Apply Bayes' theorem

Now we can find the probability that a lung cancer patient is a smoker:

P(S|C) = P(C|S) × P(S) / P(C)
P(S|C) = 0.34 × 0.11 / 0.0641
P(S|C) = 0.0374 / 0.0641
P(S|C) ≈ 0.584

Step 4 — Test for independence

Events are independent if P(S|C) = P(S). Comparing our results:

Independence test:
P(S) = 0.11 = 11%
P(S|C) ≈ 0.584 = 58.4%

Since P(S|C) ≠ P(S), smoking and lung cancer are not independent.

Solution: Method 2 — Population of 100,000 People

Sometimes it's clearer to work with actual counts rather than probabilities. Let's imagine a population of 100,000 people and trace through the numbers.

Step 1 — Break down the population by smoking status

GroupCountPercentage
Smokers11,00011%
Non-smokers89,00089%
Total100,000100%

Step 2 — Calculate cancer cases in each group

Apply the given cancer rates to each group:

GroupPopulationCancer RateCancer Cases
Smokers11,00034%11,000 × 0.34 = 3,740
Non-smokers89,0003%89,000 × 0.03 = 2,670
Total Cancer Cases100,0006.41%6,410

Step 3 — Find the smoking rate among cancer patients

Of the 6,410 people with lung cancer, 3,740 are smokers:

P(S|C) = Smokers with cancer / Total cancer cases
P(S|C) = 3,740 / 6,410 ≈ 0.584

This confirms our Bayes' theorem calculation: about 58.4% of lung cancer patients are smokers.

Answer: The probability that a lung cancer patient is a smoker is approximately 0.584 or 58.4%. Smoking and lung cancer are not independent — knowing someone has lung cancer dramatically increases the probability they're a smoker (from 11% to 58.4%).

Verification

Let's verify our answer makes sense by checking that all probabilities are consistent:

Check 1: Does P(S|C) + P(S'|C) = 1?
P(S'|C) = 1 - 0.584 = 0.416
0.584 + 0.416 = 1.000 ✓

Check 2: Alternative calculation of P(S'|C)
P(S'|C) = P(C|S') × P(S') / P(C) = 0.03 × 0.89 / 0.0641 ≈ 0.416 ✓

Check 3: Using our population table
Non-smokers with cancer: 2,670
P(S'|C) = 2,670 / 6,410 ≈ 0.416 ✓

The Power of Base Rates

This problem illustrates a crucial concept in probability: base rates matter enormously. Even though smokers are much more likely to get lung cancer (34% vs. 3%), the majority of the population doesn't smoke. This "dilutes" the smoking signal in the cancer population.

The surprising result: Even with such a strong risk factor, nearly 42% of lung cancer patients are non-smokers. This happens because non-smokers vastly outnumber smokers (89% vs. 11%), so even their low cancer rate (3%) contributes substantial numbers to the total cancer cases.

Common Pitfalls

✗ Mistake 1: Confusing P(C|S) with P(S|C)
Wrong reasoning: "34% of smokers get cancer, so 34% of cancer patients must be smokers."
Why it's wrong: This ignores base rates. The direction of conditioning matters — these are completely different questions.
✗ Mistake 2: Assuming independence means no relationship
Wrong reasoning: "Since P(S|C) ≠ P(S), there's some relationship, but we can't quantify how strong."
Why it's wrong: We can quantify it precisely. The ratio P(S|C)/P(S) = 0.584/0.11 ≈ 5.3 means lung cancer patients are about 5.3 times more likely to be smokers than the general population.
✗ Mistake 3: Forgetting the complement probability
Wrong calculation: P(C) = P(C|S) × P(S) = 0.34 × 0.11 = 0.0374
Why it's wrong: This only accounts for cancer in smokers. You must include both P(C|S) × P(S) + P(C|S') × P(S') to get the total cancer rate.

Recognizing This Problem in the Wild

You're dealing with Bayes' theorem whenever you see these patterns:

  • "Given that..." flipped around — You know P(disease|exposure) but want P(exposure|disease)
  • Medical diagnostic questions — "What's the chance this positive test means I have the disease?"
  • Base rate problems — Population percentages are given alongside conditional probabilities
  • Independence testing — "Do these two events influence each other?"
  • Survey data analysis — Large-scale population studies with breakdowns by subgroups
Key phrases to watch for: "probability that a [condition] patient has [risk factor]," "are these events independent," "given the following population data," "medical researchers know that."

What This Shows Up in Real Life

  • Medical diagnosis: Interpreting test results, understanding disease screening, calculating positive predictive values for medical tests
  • Legal evidence: Analyzing DNA matches, fingerprint evidence, and witness reliability in court cases
  • Quality control: Determining whether defective products came from specific production lines or shifts
  • Marketing analytics: Finding which customer characteristics predict buying behavior, calculating conversion rates by demographic
  • Insurance and actuarial science: Assessing risk factors for claims, setting premiums based on customer profiles

Try These Variations

1
Higher Smoking Rate
Suppose public health efforts fail and the smoking rate rises to 25%. Keeping all other probabilities the same (P(C|S) = 0.34, P(C|S') = 0.03), what is the new probability that a lung cancer patient is a smoker?
Step 1 — Update the given information

Now P(S) = 0.25 and P(S') = 0.75, while P(C|S) = 0.34 and P(C|S') = 0.03 remain the same.

Step 2 — Calculate new total cancer rate

P(C) = 0.34 × 0.25 + 0.03 × 0.75 = 0.085 + 0.0225 = 0.1075

Step 3 — Apply Bayes' theorem

P(S|C) = (0.34 × 0.25) / 0.1075 = 0.085 / 0.1075 ≈ 0.791

Verification

About 79.1% of lung cancer patients would be smokers. Higher base smoking rates dramatically increase this percentage.

2
Reverse Engineering
From the original data, we found P(S|C) ≈ 0.584. Suppose you only knew P(S) = 0.11, P(C) = 0.0641, and P(S|C) = 0.584. Could you find the original cancer risk P(C|S)?
Step 1 — Rearrange Bayes' theorem

From P(S|C) = P(C|S) × P(S) / P(C), we get:
P(C|S) = P(S|C) × P(C) / P(S)

Step 2 — Substitute known values

P(C|S) = 0.584 × 0.0641 / 0.11

Step 3 — Calculate

P(C|S) = 0.0374 / 0.11 ≈ 0.340

Verification

P(C|S) = 34%, which matches our original given value. Bayes' theorem works in both directions!

3
Three Groups
Consider three groups: heavy smokers (10%, P(C|H) = 0.45), light smokers (15%, P(C|L) = 0.18), and non-smokers (75%, P(C|N) = 0.03). What is P(H|C) — the probability a cancer patient is a heavy smoker?
Step 1 — Calculate total cancer rate

P(C) = 0.45 × 0.10 + 0.18 × 0.15 + 0.03 × 0.75
= 0.045 + 0.027 + 0.0225 = 0.0945

Step 2 — Apply Bayes' for heavy smokers

P(H|C) = P(C|H) × P(H) / P(C)
= 0.45 × 0.10 / 0.0945 ≈ 0.476

Step 3 — Calculate other groups for completeness

P(L|C) = 0.18 × 0.15 / 0.0945 ≈ 0.286
P(N|C) = 0.03 × 0.75 / 0.0945 ≈ 0.238

Verification

47.6% of cancer patients are heavy smokers. Check: 0.476 + 0.286 + 0.238 = 1.000 ✓

4
Independence Scenario
What would P(C|S') have to be for smoking and lung cancer to be independent, given P(S) = 0.11 and P(C|S) = 0.34? What would P(S|C) equal in that case?
Step 1 — Set up independence condition

For independence, we need P(S|C) = P(S) = 0.11. This means P(S'|C) = 0.89.

Step 2 — Use the independence condition

If independent, then P(C|S) = P(C|S') = P(C). Since P(C|S) = 0.34, we need P(C|S') = 0.34 too.

Step 3 — Verify with total probability

P(C) = 0.34 × 0.11 + 0.34 × 0.89 = 0.34(0.11 + 0.89) = 0.34

Answer

P(C|S') would need to be 0.34 (same as smokers), and then P(S|C) = 0.11 (same as the population rate). This would mean smoking has no effect on cancer risk.

Frequently Asked Questions

What is Bayes' theorem and when do you use it in probability? +
Bayes' theorem calculates P(A|B) when you know P(B|A). The formula is P(A|B) = P(B|A) × P(A) / P(B). Use it for 'reverse' conditional probability questions. In this problem, we know the cancer rate for smokers but want the smoking rate among cancer patients — a classic Bayes' application.
How do you test if two events are independent in probability? +
Two events A and B are independent if P(A|B) = P(A), meaning knowing B doesn't change the probability of A. Alternatively, check if P(A and B) = P(A) × P(B). Here, P(S) = 0.11 but P(S|C) ≈ 0.584, so smoking and lung cancer are strongly dependent.
Why do medical studies often report conditional probabilities backwards from what patients want to know? +
Medical studies follow large groups over time to measure disease rates among exposed vs. unexposed people — giving P(disease|exposure). But patients want P(exposure|disease) — "given I have the disease, what caused it?" Bayes' theorem bridges this gap, though it requires knowing the base rates in the population.
NJ

Dr. Neven Jurkovic

Math Professor, University of Zagreb
Expert in statistics, probability theory, and mathematical education

NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-08-23

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