Independence and Conditional Independence

Independence

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

Mathematically: $P(A \cap B) = P(A) \times P(B)$

Two events $A$ and $B$ might be dependent, but become independent once we know a third event $C$ .

Mathematically: $P(A \cap B | C) = P(A|C) \times P(B|C)$

A standard deck of 52 cards is well-shuffled. Let Event $A$ be drawing a Heart. Let Event $B$ be drawing a King. Are events $A$ and $B$ independent?

View Step-by-Step Solution

To check independence, we must verify if $P(A \cap B) = P(A) \times P(B)$ .

$P(A) = 13/52 = 1/4$ (There are 13 Hearts)
$P(B) = 4/52 = 1/13$ (There are 4 Kings)
$P(A \cap B)$ is the probability of drawing the King of Hearts. There is exactly 1 King of Hearts, so $P(A \cap B) = 1/52$ .

Check the math: $(1/4) \times (1/13) = 1/52$ . Since $1/52 = 1/52$ , the events are independent.