The following definition provides an intuitive definition of the concept of independence for two events, and then we look at an example that provides a computational way for determining when events are independent.

### Example \(\PageIndex{1}\)

Suppose that events \(A\) and \(B\) are independent. We rewrite Equations \ref{indep} using the definition of conditional probability:

$$P(A\ |\ B) = P(A) \quad \Rightarrow\quad \frac{P(A\cap B)}{P(B)} = P(A) \notag$$

$$ \text{and} \notag$$

$$P(B\ |\ A) = P(B) \quad \Rightarrow\quad \frac{P(A\cap B)}{P(A)} = P(B) \notag$$

In each of the expressions on the right-hand side above we isolate \(P(A\cap B)\):

$$\frac{P(A\cap B)}{P(B)} = P(A) \quad \Rightarrow\quad P(A\cap B) = P(A)P(B) \notag$$

$$ \text{and} \notag$$

$$\frac{P(A\cap B)}{P(A)} = P(B) \quad \Rightarrow\quad P(A\cap B) = P(A)P(B) \notag$$

Both expressions result in \(P(A\cap B) = P(A)P(B)\). Thus, we have shown that if events \(A\) and \(B\) are independent, then the probability of their intersection is equal to the product of their individual probabilities. We state this fact in the next definition.

### Definition \(\PageIndex{2}\)

Events \(A\) and \(B\) are **independent** if $$P(A\cap B) = P(A)P(B).\notag$$

###

Generally speaking, Definition 2.3.2 tends to be an easier condition than Definition 2.3.1 to verify when checking whether two events are independent.

### Example \(\PageIndex{2}\)

Consider the context of Exercise 2.2.1, where we randomly draw a card from a standard deck of 52 and \(C\) denotes the event of drawing a club, \(K\) the event of drawing a King, and \(B\) the event of drawing a black card.

Are \(C\) and \(K\) independent events? Recall that \(P(C\cap K) = 1/52\), and note that \(P(C) = 13/52\) and \(P(K) = 4/52\). Thus, we have

$$P(C\cap K) = \frac{1}{52} = P(C)P(K) = \frac{13}{52}\times\frac{4}{52},\notag$$

indicating that \(C\) and \(K\) **are independent**.

Are \(C\) and \(B\) independent events? Recall that \(P(C\cap B) = 13/52\), and note that \(P(B) = 26/52\). Thus, we have

$$P(C\cap B) = \frac{13}{52} \neq P(C)P(B) = \frac{13}{52}\times\frac{26}{52},\notag$$

indicating that \(C\) and \(B\) **are ***not *independent.

Let's think about the results of this example intuitively. To say that \(C\) and \(K\) are independent means that knowing that one of the events occurs does not affect the probability of the other event occurring. In other words, knowing that the card drawn is a King does not influence the probability of the card being a club. The proportion of clubs in the entire deck of 52 is the same as the proportion of clubs in just the collection of Kings: \(1/4\). On the other hand, \(C\) and \(B\) are not independent (AKA *dependent*) because knowing that the card drawn is club indicates that the card *must be black*, i.e., the probability that the card is black is 1. Alternately, knowing that the card drawn is black increases the probability that the card is a club, since the proportion of clubs in the entire deck is \(1/4\), but the proportion of clubs in the collection of black cards is \(1/2\).