Probability with extra information
Conditional probability is the probability of an event when you already know something else has happened.
P(A | B) = probability of A given B.
Formula: P(A | B) = P(A and B) / P(B)
Example with cards
Draw one card. What’s P(king | face card)?
- Face cards: 12 (J, Q, K in 4 suits)
- Kings among them: 4
- P = 4/12 = 1/3
Without the condition, P(king) = 4/52 = 1/13. Knowing it’s a face card changes the probability.
Using a two-way table
Survey of 200 students:
| Plays sport | No sport | Total | |
|---|---|---|---|
| Grade 11 | 60 | 40 | 100 |
| Grade 12 | 45 | 55 | 100 |
| Total | 105 | 95 | 200 |
- P(plays sport | Grade 11) = 60/100 = 0.60
- P(Grade 11 | plays sport) = 60/105 ≈ 0.571
Notice: P(A | B) and P(B | A) are usually different!
The multiplication rule
Rearranging gives the multiplication rule:
P(A and B) = P(A) × P(B | A)
Example: Draw two cards without replacement. P(both hearts)?
13/52 × 12/51 = 1/17 ≈ 0.059
Independence
Events are independent if knowing one doesn’t change the other:
P(A | B) = P(A)
From the table: P(sport) = 0.525 but P(sport | Grade 11) = 0.60, so grade and sport are not independent.
Bayes’ theorem
Bayes’ theorem reverses a conditional probability:
P(A | B) = P(B | A) × P(A) / P(B)
Example: medical testing
A disease affects 1% of people. A test detects it 95% of the time when present, but gives a false positive 5% of the time when absent. If someone tests positive, what’s the probability they have the disease?
Imagine 10,000 people:
| Test + | Test − | Total | |
|---|---|---|---|
| Has disease | 95 | 5 | 100 |
| No disease | 495 | 9,405 | 9,900 |
| Total | 590 | 9,410 | 10,000 |
P(disease | positive) = 95/590 ≈ 16%
Surprisingly low! Because the disease is rare, most positives are false positives. This is why doctors confirm results with further tests.
Worked examples
Example 1: Roll a die. P(6 | even)? 1/3.
Example 2: P(A and B) = 0.12, P(B) = 0.4. P(A | B)? 0.3.
Example 3: From the table, P(no sport | Grade 12)? 55/100 = 0.55.
Common mistakes
- Confusing P(A | B) with P(B | A).
- Using the whole total instead of the "given" group as the denominator.
- Ignoring base rates (as in the medical test example).
Try it yourself
- Draw a card. P(red | heart)?
- P(ace | spade)?
- P(A) = 0.5, P(B | A) = 0.3. P(A and B)?
- Using the student table, P(Grade 12 | no sport)?
Answers: 1) 1 2) 1/13 3) 0.15 4) 55/95 ≈ 0.579
Quick recap
- Conditional probability P(A | B) = P(A and B)/P(B).
- Two-way tables make conditional probabilities clear.
- Bayes’ theorem reverses conditions; base rates matter.