Equivalently (any of these):

• $P(A|B) = P(A)$
• $P(B|A) = P(B)$
• $P(A ∩ B) = P(A) · P(B)$

Multiplication Rule for Independent Events

When $A$ and $B$ independent:

$P(A \cap B) = P(A) · P(B)$

Example: Flip two fair coins

• $P(\text{Coin 1 is heads}) = 0.5$
• $P(\text{Coin 2 is heads}) = 0.5$
• $P(\text{Both heads}) = 0.5 × 0.5 = 0.25$

Testing for Independence

Method: Check if $P(A \cap B) = P(A) · P(B)$

Example with data:

• $P(\text{rain}) = 0.3$
• $P(\text{pain}) = 0.4$
• $P(\text{rain and pain}) = 0.11$

Check: $P(\text{rain}) · P(\text{pain}) = 0.3 × 0.4 = 0.12$

Since $0.11 \neq 0.12$, not perfectly independent (likely due to sampling variation, but close)

Common Real-World Examples

Independent:

• Coin flips (fair coin)
• Die rolls (fair die)
• Card draws with replacement
• Gender and eye color (biological context)

Dependent:

• Card draws without replacement
• Weather and joint pain (claims lack strong evidence)
• ACT score and college GPA (both reflect ability)
• Smoking and lung cancer (causation, strong dependence)

Worked Example

Scenario:

• $P(\text{Student studies}) = 0.8$
• $P(\text{Student passes}) = 0.9$
• $P(\text{Studies and passes}) = 0.76$

Are studying and passing independent?

Check: $0.8 × 0.9 = 0.72 \neq 0.76$

Conclusion: Dependent (studying increases probability of passing)

$P(\text{Pass} | \text{Study}) = \frac{0.76}{0.80} = 0.95 > 0.9 = P(\text{Pass})$

Common Mistakes

• Confusing independent with mutually exclusive (opposite concepts!)
• Assuming real-world events independent without testing
• Using multiplication rule when events are dependent

Decision Rule

Are events independent? → Use $P(A ∩ B) = P(A) · P(B)$ Are events mutually exclusive? → Use $P(A ∩ B) = 0$

AP Exam Tip

"Are A and B independent?" requires checking the definition: does knowing B change the probability of A? If yes, dependent.