Independent Events

In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event. In other words, knowing that one event has occurred does not change the likelihood of the other event occurring.

For two events A and B, they are independent if and only if the probability of both events occurring together (the intersection of A and B) is equal to the product of their individual probabilities. Mathematically, this is expressed as:

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

If this equation holds true, then $A$ and $B$ are independent events.

For example, consider the events of flipping a coin and rolling a die. Let event $A$ be getting heads on the coin flip, and event $B$ be rolling a 4 on the die. The probability of getting heads on the coin is $0.5$, and the probability of rolling a 4 on the die is $\frac{1}{6}$​. Since these two events do not influence each other, they are independent. Therefore, the probability of both events happening together is:

$P(A \cap B) = P(A) \times P(B) = 0.5 \times \frac{1}{6} = \frac{1}{12}$

This confirms that the events are independent because the probability of their intersection is equal to the product of their individual probabilities.