Addition Law (or Law of Total Probability): The Addition Law is a fundamental principle in probability theory that allows us to calculate the probability of an event by considering the probabilities of its mutually exclusive and exhaustive components. It states that if A and B are two mutually exclusive events (meaning they cannot occur simultaneously), then the probability of either event A or event B occurring is the sum of their individual probabilities. Mathematically, it can be expressed as: P(A ∪ B) = P(A) + P(B)

The Addition Law can be extended to more than two events. If A1, A2, …, An are mutually exclusive events, then the probability of any of these events occurring is given by: P(A1 ∪ A2 ∪ … ∪ An) = P(A1) + P(A2) + … + P(An)

Multiplication Law (or Multiplication Rule): The Multiplication Law is another important principle in probability theory that allows us to calculate the probability of the joint occurrence of two or more events. It states that if A and B are two independent events, then the probability of both events A and B occurring is the product of their individual probabilities. Mathematically, it can be expressed as: P(A ∩ B) = P(A) * P(B)

Similarly, the Multiplication Law can be extended to more than two independent events. If A1, A2, …, An are independent events, then the probability of their joint occurrence is given by: P(A1 ∩ A2 ∩ … ∩ An) = P(A1) * P(A2) * … * P(An)

Bayes’ Theorem: Bayes’ theorem is a fundamental result in probability theory that relates conditional probabilities. It allows us to update the probability of an event based on new information or evidence. Bayes’ theorem states that if A and B are two events, and P(A) and P(B) are the probabilities of A and B, respectively, then the conditional probability of event A given that event B has occurred (P(A | B)) can be calculated as: P(A | B) = (P(B | A) * P(A)) / P(B)

In this formula, P(B | A) represents the conditional probability of event B given that event A has occurred. P(A) and P(B) are the individual probabilities of events A and B, respectively. P(B) ≠ 0, meaning that the probability of event B must be non-zero.

Bayes’ theorem is particularly useful in situations where we have prior knowledge about the probabilities of events and want to update our beliefs based on new evidence or observations. It has applications in various fields, including medical diagnosis, machine learning, and data analysis.