Probability when we have additional information about an event
For any two events A and B of a sample space, with P(B) ≠ 0:
Problem: A card is drawn from a deck. Given that it's a red card, what's the probability it's a heart?
Solution:
Let A = Heart, B = Red card
Total hearts = 13, Total red cards = 26, Red hearts = 13
P(A ∩ B) = 13/52
P(B) = 26/52
P(A|B) = (13/52) ÷ (26/52) = 13/26 = 1/2
Problem: Two dice are rolled. Given that the sum is greater than 7, find the probability that both dice show odd numbers.
Solution:
Event A: Both dice show odd = {(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)}
Event B: Sum > 7 = {(1,7)...wait, max is 6} = many pairs
A ∩ B (both odd AND sum > 7) = {(3,5), (5,3), (5,5)}
P(A|B) = n(A ∩ B) / n(B) = 3/15 = 1/5
Bayes' Theorem relates the conditional probabilities of two events. It allows us to find the probability of the cause given the effect.
If A₁, A₂, ..., Aₙ are mutually exclusive and exhaustive events:
Problem: A disease affects 1% of the population. A test has 95% accuracy for both positive and negative cases. If a person tests positive, what's the probability they have the disease?
Solution:
Let D = Has disease, + = Tests positive
P(D) = 0.01, P(+|D) = 0.95, P(+|D') = 0.05
P(+) = P(+|D)P(D) + P(+|D')P(D') = 0.95(0.01) + 0.05(0.99) = 0.0095 + 0.0495 = 0.059
P(D|+) = [P(+|D)P(D)] / P(+) = (0.95 × 0.01) / 0.059 = 0.0095 / 0.059 ≈ 0.161 or 16.1%
Problem: Factory A produces 60% of items, Factory B produces 40%. 2% of A's items are defective, 3% of B's are defective. An item is found defective. What's the probability it came from Factory A?
Solution:
P(A) = 0.6, P(B) = 0.4, P(D|A) = 0.02, P(D|B) = 0.03
P(D) = P(D|A)P(A) + P(D|B)P(B) = 0.02(0.6) + 0.03(0.4) = 0.012 + 0.012 = 0.024
P(A|D) = [P(D|A)P(A)] / P(D) = (0.02 × 0.6) / 0.024 = 0.012 / 0.024 = 0.5 or 50%
Q: Define conditional probability and write its formula
Q: Apply conditional probability to practical scenarios
Q: Use Bayes' Theorem to find posterior probability
Common Mistake: Confusing P(A|B) with P(B|A). Always be clear about what is given!
Helpful Tip: Draw a Venn diagram or tree diagram to visualize the problem
Pattern Recognition: Bayes' Theorem appears frequently in combination with probability distributions
Strategic Approach: For multi-stage problems, use tree diagrams systematically
Common Trap: Assuming independence when events are actually dependent - always check!
Real-world Applications: Bayes' Theorem is foundational in statistics and machine learning - understanding it deeply helps in advanced questions