Final answer:
Conditional probability, represented as P(A|B), is the likelihood of event A occurring after event B has occurred. The calculations hingе on whether events are independent, mutually exclusive, or interdependent, with the formula adapted accordingly.
Step-by-step explanation:
Conditional probability, denoted as P(A|B), is the probability of event A occurring given that event B has already happened. This concept is a fundamental aspect of probability theory and is often encountered in various mathematical scenarios. To calculate conditional probability, you use the formula P(A|B) = P(A AND B) / P(B), assuming that P(B) > 0. When applying this concept, it is crucial to understand whether the events are independent, mutually exclusive, or neither.
For independent events, the probability of both events A and B occurring, known as the joint probability, can be found using the multiplication rule: P(A AND B) = P(A)P(B). However, if the events are not independent, the multiplication rule is adjusted to account for the dependency: P(A AND B) = P(A|B)P(B). Bayesian statistics also use an approach to conditional probability, involving the Bayesian theorem, which is particularly useful in updating the probability estimate for a hypothesis as additional evidence is acquired.