To determine if two events are independent in probability, you use the definition of independence:

Two events $A$ and $B$ are independent if:

$P(A \cap B) = P(A) \cdot P(B)$ This means the probability of both events happening together is equal to the product of their individual probabilities.

Steps to Check for Independence:

1. Find $P(A)$: Calculate the probability of event $A$.
2. Find $P(B)$: Calculate the probability of event $B$.
3. Find $P(A \cap B)$: Calculate the probability that both $A$ and $B$ happen simultaneously.
4. Compare $P(A \cap B)$ and $P(A) \cdot P(B)$:
   • If they are equal, $A$ and $B$ are independent.
   • If they are not equal, $A$ and $B$ are dependent.

Additional Notes:

• Mutually Exclusive Events: If $A$ and $B$ are mutually exclusive (they cannot occur together), then $P(A \cap B) = 0$. Such events are never independent unless one or both have zero probability.
• Independence is about the relationship between probabilities, not whether the events occur simultaneously.

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Example:

Suppose:

• $P(A) = 0.4$
• $P(B) = 0.5$
• $P(A \cap B) = 0.2$

Check: $P(A) \cdot P(B) = 0.4 \cdot 0.5 = 0.2$ Since $P(A \cap B) = 0.2$, the events $A$ and $B$ are independent.

Would you like further clarification or examples?

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5 Related Questions:

1. What is the difference between independent and mutually exclusive events?
2. How do you calculate conditional probability, and how does it relate to independence?
3. Can three or more events be mutually independent? How do you test for that?
4. What are examples of independent events in real-life situations?
5. What happens when one event has a probability of zero? Can the events still be independent?

Tip:

Always verify independence using the mathematical formula, as intuition about whether events are independent