Independence

What is Independence?

Independent Events: Occurrence of one event doesn't affect probability of the other

Formal definition: Events A and B are independent if:

$P(A|B) = P(A)$

Equivalently:

• P(B|A) = P(B)
• P(A ∩ B) = P(A) × P(B)

Key insight: Knowing one occurred gives no information about the other

Testing for Independence

Method 1: Conditional Probability

Check if P(A|B) = P(A)

Example: Flip coin twice

• P(H on flip 2) = 1/2
• P(H on flip 2|H on flip 1) = 1/2
• Equal → Independent

Method 2: Multiplication Rule

Check if P(A and B) = P(A) × P(B)

Example: Roll die, flip coin

• P(6 on die) = 1/6
• P(H on coin) = 1/2
• P(6 and H) = 1/12 = 1/6 × 1/2 ✓
• Independent

Method 3: Two-Way Table

For independence, each cell should equal (row total × column total)/grand total

Independence vs Mutually Exclusive

IMPORTANT: Independent ≠ Mutually Exclusive

Mutually Exclusive: Can't both occur (P(A ∩ B) = 0)
Independent: One doesn't affect other (P(A ∩ B) = P(A) × P(B))

In fact: If P(A) > 0 and P(B) > 0, mutually exclusive events are DEPENDENT!

Why? If A occurs, B definitely can't occur, so P(B|A) = 0 ≠ P(B)

Example:

• A: Roll 2 on die
• B: Roll 5 on die
• Mutually exclusive (can't both happen)
• NOT independent (if A occurs, B can't, so they're dependent)

Independence in Practice

Sampling with replacement: Draws are independent
Sampling without replacement: Draws are dependent

Example: Two cards from deck

With replacement:

• P(First ace) = 4/52
• P(Second ace|First ace) = 4/52 (replaced first card)
• P(Second ace) = 4/52
• Independent ✓

Without replacement:

• P(First ace) = 4/52
• P(Second ace|First ace) = 3/51
• P(Second ace) ≈ 4/52 (overall, across all possible first cards)
• NOT independent (but close if sample is tiny compared to population)

10% Condition

Rule of thumb: If sample size < 10% of population, treat as independent even without replacement

Why? Removing small fraction doesn't appreciably change probabilities

Example: 5 cards from deck

• 5/52 ≈ 9.6% < 10%
• Can approximate as independent (slight error)

Example: 20 cards from deck

• 20/52 ≈ 38% > 10%
• Must account for dependence

Multiplication Rule for Independent Events

If A and B are independent:

$P(A \text{ and } B) = P(A) \times P(B)$

Extends to multiple events:

$P(A_1 \text{ and } A_2 \text{ and } ... \text{ and } A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)$

Example: Flip coin 3 times, P(HHH) = 1/2 × 1/2 × 1/2 = 1/8

At Least One Calculations

"At least one" problems: Often easier to use complement

P(At least one A) = 1 - P(No A)

Example: Flip coin 3 times, find P(at least one head)

Long way: P(1H) + P(2H) + P(3H) = complicated

Short way: P(At least 1H) = 1 - P(No H) = 1 - P(TTT) = 1 - (1/2)³ = 1 - 1/8 = 7/8

Example: Shoot basketball, 70% success rate, 3 attempts

P(Make at least one) = 1 - P(Miss all 3) = 1 - (0.3)³ = 1 - 0.027 = 0.973

Checking Independence from Two-Way Table

100 students:

| | Male | Female | Total | |-----------|------|--------|-------| | Athlete | 24 | 16 | 40 | | Non-athlete| 36 | 24 | 60 | | Total | 60 | 40 | 100 |

Check independence of Athlete and Male:

Method 1:

• P(Athlete) = 40/100 = 0.4
• P(Athlete|Male) = 24/60 = 0.4
• Equal → Independent ✓

Method 2:

• P(Athlete and Male) = 24/100 = 0.24
• P(Athlete) × P(Male) = (40/100) × (60/100) = 0.4 × 0.6 = 0.24
• Equal → Independent ✓

Method 3: Expected cell count

• Expected = (row total × column total)/grand total = (40 × 60)/100 = 24
• Actual = 24
• Equal → Independent ✓

Real-World Independence

Independent (usually):

• Coin flips
• Die rolls
• Different people's responses (if random sample)
• Successive free throws (debatable!)
• Rain in New York and LA on same day

Dependent:

• Cards without replacement
• Success/failure of teammates
• Weather on consecutive days
• Stock prices over time
• Contagious disease among contacts

Independence of Complements

If A and B are independent:

• A and B^c are independent
• A^c and B are independent
• A^c and B^c are independent

Example: Two independent coin flips

• If flips are independent, then "First heads" and "Second tails" are also independent

Common Mistakes

❌ Assuming events are independent without checking
❌ Thinking mutually exclusive means independent (opposite!)
❌ Forgetting 10% condition for sampling
❌ Using multiplication rule when events aren't independent
❌ Confusing P(A and B) with P(A) + P(B)

Practice Strategy

1. Question: Are events independent?
2. Test: Check if P(A|B) = P(A) or P(A and B) = P(A) × P(B)
3. Context: Does it make sense? (Replacement? Separate processes?)
4. Calculate: Use appropriate rule (multiply if independent)

Quick Reference

Definition: P(A|B) = P(A) or P(B|A) = P(B)

Multiplication: P(A and B) = P(A) × P(B) if independent

Complement: P(At least 1) = 1 - P(None)

10% Rule: Treat as independent if n < 0.10N

Key: Independent ≠ Mutually Exclusive

Remember: Independence means events don't influence each other. Always verify independence before using multiplication rule!