Additive and Multiplicative Rules of probability

Addition and Multiplication Theorems in Probability

Addition Theorem on Probability:

• Definition: If A and B are any two events, then the probability of either A or B occurring (or both) is given by P(A∪B)=P(A)+P(B)−P(A∩B).
• Proof: Derived from set theory principles, where P(A∪B) represents the probability of the union of events A and B.
• Example: If $P(A)=\; 1/2$ and P(B)=1/3 ​, then

Multiplication Theorem on Probability:

• Definition: If A and B are any two events with non-zero probabilities, then P(A∩B)=P(A)⋅P(B∣A)=P(B)⋅P(A∣B).
• Interpretation: P(B∣A) and P(A∣B)are conditional probabilities, representing the probability of B given A and A given B, respectively.
• Example: If P(A)=1/5 and P(B∣A)=1/3 ​, then P(A∩B)=1/5⋅1/3=1/15.

Independent Events:

• Definition: Events A and B are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, P(A∩B)=P(A)⋅P(B)
• Example: Drawing a diamond and drawing an ace from a deck of cards are independent events if P(diamond)=1/4 and P(ace)=1/13​, with P(diamond ∩ ace) = 1/52​.

Additional Notes:

• General Formula for Union of Events: P(A∪B) = P(A)+P(B)−P(A∩B) is extended to more than two events using inclusion-exclusion principles.
• Mutually Exclusive Events: If A and B are mutually exclusive (no overlap), P(A∩B)=0, simplifying to P(A∪B) = P(A)+P(B).
• Conditional Probability: P(B∣A)and P(A∣B)are crucial in assessing probabilities when one event's occurrence depends on another's.

These theorems and concepts form the foundation of probabilistic reasoning, essential in various fields such as statistics, economics, and decision theory. They provide tools to calculate and understand the likelihood of events based on given information and assumptions.