01 Key Concepts
The Core Idea
Bayes' Theorem updates the probability of a cause (A) after observing some evidence (B).
Bayes' Theorem Formula
P(A|B) = [P(B|A) * P(A)] / P(B).
Law of Total Probability
When A has several possible non-overlapping cases (A1, A2, ...), P(B) = sum of P(B|Ai)*P(Ai) over all i — needed to compute the denominator in Bayes' formula.
Prior vs. Posterior Probability
P(A) is the 'prior' belief before seeing evidence. P(A|B) is the 'posterior' belief after incorporating the evidence B.
Common Application
Medical testing: given a positive test result (B), find the true probability of having a disease (A), accounting for false positives and the disease's rarity.
02 Key Formulas
- P(A|B) = [P(B|A) * P(A)] / P(B)
- P(B) = P(B|A)*P(A) + P(B|A')*P(A')
03 Solved Examples
Example 1 A disease affects 1% of a population. A test correctly detects it 90% of the time (true positive) and gives a false positive 5% of the time. Find P(has disease | tests positive).
- Let A = has disease, so P(A) = 0.01, P(A') = 0.99.
- P(positive|A) = 0.90, P(positive|A') = 0.05.
- Find P(positive) = P(positive|A)*P(A) + P(positive|A')*P(A') = 0.90*0.01 + 0.05*0.99 = 0.009 + 0.0495 = 0.0585.
- Apply Bayes: P(A|positive) = (0.90*0.01)/0.0585 = 0.009/0.0585.
Answer: Approximately 0.154, or about 15.4%
Example 2 Two factories supply light bulbs: Factory A makes 60% of bulbs with a 2% defect rate; Factory B makes 40% with a 5% defect rate. A random bulb is defective. Find P(it came from Factory B).
- P(A)=0.6, P(B)=0.4. P(defective|A)=0.02, P(defective|B)=0.05.
- P(defective) = 0.02*0.6 + 0.05*0.4 = 0.012 + 0.02 = 0.032.
- P(B|defective) = (0.05*0.4)/0.032 = 0.02/0.032.
Answer: 0.625, or 62.5%
Example 3 Explain in words what P(A|B) represents versus P(A) in Bayes' Theorem.
- P(A) is the prior probability — our belief about A before any evidence.
- P(A|B) is the posterior probability — the updated belief about A after learning that B occurred.
Answer: P(A) is the prior; P(A|B) is the updated (posterior) probability after observing evidence B
04 Practice Questions
1If P(A)=0.02, P(B|A)=0.8, P(B)=0.1, find P(A|B).
0.16
2A disease affects 2% of people; a test is 95% accurate for both positives and negatives. Set up P(B) using the law of total probability (do not solve fully).
P(B)=0.95*0.02 + 0.05*0.98
3Solve the previous P(B).
0.019 + 0.049 = 0.068
4What is a 'prior probability' in Bayes' Theorem?
The initial probability of an event before new evidence is considered
5What is a 'posterior probability' in Bayes' Theorem?
The updated probability after accounting for new evidence
📄 Bayes' Theorem — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.