To demonstrate that if events A and B are independent, then A and B are also independent, we first need to define what it means for two events to be independent.
Two events A and B are said to be independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this is expressed as:
- P(A and B) = P(A) * P(B)
Now, let’s explore this in a more structured way:
Step 1: Understand the Independence Definition
According to the definition, for A and B to be independent, the joint probability of both A and B occurring must equal the product of their individual probabilities.
Step 2: Consider Independence of Events
When we say A and B are independent, we mean:
- P(A and B) = P(A) * P(B)
Step 3: Explore the Implication of Independence
From the definition, if events A and B are independent, we can infer the following:
- The probability of A happening does not change if B occurs, and vice versa.
- Thus, we can also write P(B|A) = P(B) and P(A|B) = P(A).
Step 4: Show that A and B are Independent Based on the Independence Premise
Now, we will prove that if A and B are independent, then they maintain that independence:
- By the definition of independence, the relationship holds: P(A and B) = P(A) * P(B).
- The joint probability is equal to the individual probabilities multiplied together, reaffirming their independence.
Conclusion
Thus, the statement is verified: If A and B are independent events, then indeed A and B are also independent in every sense within the context of probability theory.
In summary, proving independence between events involves understanding how their interactions govern their relationship, and reinforcing that understanding with statistical definitions solidifies the concept of independence.