How can independence between two events be tested using probabilities?

In probability and statistics, two events $A$ and $B$ are considered independent if the occurrence of one does not change the probability of the other occurring. You can test for independence using two primary mathematical approaches:123

• Conditional Probability: Events $A$ and $B$ are independent if the conditional probability of $B$ given $A$ is equal to the marginal probability of $B$. This is expressed as:
  $$P(B|A)=P(B)$$
  If this equality holds, knowing that $A$ has occurred provides no information about the likelihood of $B$.23

• Joint Probability: Alternatively, events are independent if the probability of their joint occurrence (the intersection of $A$ and $B$) is equal to the product of their individual probabilities:
  $$P(A\cap B)=P(A)\times P(B)$$
  When this condition is met, the events are independent, meaning their joint behavior is simply the product of their separate behaviors.1

If these conditions are not met, the events are considered dependent, meaning the outcome of one influences the probability of the other.13