Two events A and B are said to be independent if:

• A and B are mutually exclusive
• P(A')P(B') = [1 - P(A)][1 - P(B)]
• P(A) = P(B)
• P(A) + P(B) = 1

Two events A and B are considered independent when the occurrence of one does not affect the probability of the other happening. This means that knowing whether event A has occurred gives no information about the likelihood of event B occurring, and vice versa.

Key Characteristics of Independent Events

• Definition: Events A and B are independent if P(A and B) = P(A) * P(B).
• Mutual Exclusivity: Independent events are not mutually exclusive; they can occur at the same time.
• Complementary Probabilities: The relationship P(A')P(B') = [1 - P(A)][1 - P(B)] holds true for independent events.

Understanding Independence

For example, if you flip a coin and roll a die, the outcome of the coin flip (heads or tails) does not influence the result of the die roll (1 through 6). Thus, these two events are independent.

Important Note

It’s crucial to differentiate between independent events and mutually exclusive events. If A and B are mutually exclusive, the occurrence of one event means the other cannot happen, which is not the case for independent events.