Question
What does it mean for two events to be independent in probability?
Ask by Cook Peterson. in India
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Two events are independent in probability if the occurrence of one does not affect the probability of the other.
Solution
In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event.
Mathematically, two events \( A \) and \( B \) are independent if the following condition holds:
\[
P(A \cap B) = P(A) \cdot P(B)
\]
Where:
- \( P(A \cap B) \) is the probability that both events \( A \) and \( B \) occur.
- \( P(A) \) is the probability that event \( A \) occurs.
- \( P(B) \) is the probability that event \( B \) occurs.
If this equation is true, it indicates that knowing whether event \( A \) has occurred does not provide any information about whether event \( B \) has occurred, and vice versa.
In simpler terms, if two events are independent, the outcome of one event does not influence the outcome of the other event.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
For two events to be independent in probability means that the occurrence of one event does not affect the probability of the other event occurring. Simply put, if knowing that event A occurred gives no information about the likelihood of event B occurring, then they are considered independent. Mathematically, this is expressed as P(A and B) = P(A) * P(B). Think of it like tossing a coin and rolling a die at the same time! The result of the coin toss—whether it lands on heads or tails—has no impact on what number you roll on the die. So, toss away without worry; those events won't rain on each other's parade!
