key term - P(A ∪ B)

Definition

P(A ∪ B) represents the probability of the union of events A and B. The union of two events refers to the occurrence of either event A, event B, or both events A and B. It is the sum of the individual probabilities of the two events, minus the probability of their intersection.

5 Must Know Facts For Your Next Test

1. The formula for the probability of the union of two events, A and B, is: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
2. If events A and B are mutually exclusive, then P(A ∩ B) = 0, and the formula simplifies to: P(A ∪ B) = P(A) + P(B).
3. If events A and B are independent, then P(A ∩ B) = P(A) × P(B), and the formula becomes: P(A ∪ B) = P(A) + P(B) - P(A) × P(B).
4. The probability of the union of events A and B can never be greater than the sum of their individual probabilities: P(A ∪ B) ≤ P(A) + P(B).
5. Understanding the concept of P(A ∪ B) is crucial in calculating the probability of combined or alternative events in various statistical and probability problems.

Review Questions

• Explain the relationship between P(A ∪ B) and the probabilities of events A and B.
  • The probability of the union of events A and B, P(A ∪ B), is the sum of the individual probabilities of events A and B, minus the probability of their intersection, P(A ∩ B). This can be expressed as the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This relationship is important in understanding the combined likelihood of two events occurring, especially when they are mutually exclusive or independent.
• Describe how the formula for P(A ∪ B) changes when the events A and B are mutually exclusive or independent.
  • When events A and B are mutually exclusive, meaning they cannot occur simultaneously, the probability of their intersection P(A ∩ B) is 0. In this case, the formula for P(A ∪ B) simplifies to: P(A ∪ B) = P(A) + P(B). However, if events A and B are independent, meaning the occurrence of one does not affect the probability of the other, then P(A ∩ B) = P(A) × P(B), and the formula becomes: P(A ∪ B) = P(A) + P(B) - P(A) × P(B).
• Explain the significance of the inequality P(A ∪ B) ≤ P(A) + P(B) in the context of probability.
  • The inequality P(A ∪ B) ≤ P(A) + P(B) is an important principle in probability theory. It states that the probability of the union of two events, A and B, can never be greater than the sum of their individual probabilities. This is because the union of events A and B includes the possibility of either event occurring, but it does not double-count the probability of their intersection. This inequality helps ensure that probability calculations remain within the valid range of 0 to 1 and provides a useful constraint in solving various probability problems.

Related terms

Probability:

The measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

Mutually Exclusive Events:

Events that cannot occur simultaneously, meaning the occurrence of one event precludes the occurrence of the other.

Independent Events:

Events where the occurrence of one event does not affect the probability of the other event occurring.