Independence

Definition

Two events $A$ and $B$ are independent if knowing that one occurred does not change the probability of the other:

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

Equivalently: $P(A \cap B) = P(A) \cdot P(B)$

Testing for Independence

To check if events are independent, verify one of these:

1. $P(A|B) = P(A)$
2. $P(B|A) = P(B)$
3. $P(A \cap B) = P(A) \cdot P(B)$

If any one of these holds, all three hold.

Example with Two-Way Table

| | Pass | Fail | Total | |---|---|---|---| | Studied | 72 | 8 | 80 | | Didn't Study | 12 | 8 | 20 | | Total | 84 | 16 | 100 |

Check: Is passing independent of studying?

• $P(\text{Pass}) = 84/100 = 0.84$
• $P(\text{Pass} | \text{Studied}) = 72/80 = 0.90$
• Since $0.90 \neq 0.84$, the events are not independent

Independence vs. Mutually Exclusive

These are different concepts:

• Mutually exclusive: $P(A \cap B) = 0$ — events cannot both occur
• Independent: $P(A \cap B) = P(A) \cdot P(B)$ — events don't affect each other

If $P(A) > 0$ and $P(B) > 0$:

• Mutually exclusive events are never independent
• Independent events are never mutually exclusive

Independent Trials

When sampling with replacement or from a very large population, successive selections are independent.

Rule of thumb: Selections are approximately independent if the sample is less than 10% of the population (the 10% condition).

AP Tip: Don't confuse "independent events" with "independent variable" — they are different concepts in statistics.