Conditional Probability and Independence

By the end of this lesson, you will be able to calculate conditional probabilities, determine whether two events are independent, and use tree diagrams to organize probability calculations.

Conditional Probability

Conditional probability: the probability that event A occurs given that event B has already occurred (or is known to be true).

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{(provided } P(B) > 0\text{)}$$

Read "$P(A \mid B)$" as "the probability of A given B."

Intuition: Given that B has occurred, the sample space is reduced to just the outcomes in B. The conditional probability is the fraction of B's outcomes that are also in A.

Example: A fair card is drawn from a standard deck. Given that the card is a face card (J, Q, K), what is the probability it is a King?

$P(\text{King} \mid \text{face card}) = \frac{P(\text{King AND face card})}{P(\text{face card})} = \frac{4/52}{12/52} = \frac{4}{12} = \frac{1}{3}$

Two-Way Tables and Conditional Probability

Two-way tables are extremely useful for computing conditional probabilities.

Example:

	Takes AP Statistics	Doesn't take AP Stats	Total
Intends college	80	120	200
Doesn't intend college	20	80	100
Total	100	200	300

$P(\text{AP Stats} \mid \text{intends college}) = \frac{80}{200} = 0.40$

$P(\text{AP Stats} \mid \text{doesn't intend college}) = \frac{20}{100} = 0.20$

Since these are different, taking AP Statistics and intending college are associated (not independent).

Independence

Events A and B are independent if knowing that B occurred does not change the probability that A occurred.

Definition: A and B are independent if and only if: $$P(A \mid B) = P(A)$$

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

This second form is often called the .