Math Problem Statement

Given that P(A or B) = , P(A) = , and P(A and B) = , find P(B). Express the probability as a simplified fraction.

Solution

To find P(B)P(B), we can use the formula for the probability of the union of two events AA and BB:

P(A or B)=P(A)+P(B)P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Given:

  • P(A or B)P(A \text{ or } B)
  • P(A)P(A)
  • P(A and B)P(A \text{ and } B)

We can rearrange this formula to solve for P(B)P(B):

P(B)=P(A or B)P(A)+P(A and B)P(B) = P(A \text{ or } B) - P(A) + P(A \text{ and } B)

Plug in the values once provided, and then simplify the result.

Please provide the values for P(A or B)P(A \text{ or } B), P(A)P(A), and P(A and B)P(A \text{ and } B) so we can continue.

Would you like more details about this process?

Here are 5 related questions:

  1. How is the union of probabilities different from their intersection?
  2. Why do we subtract P(A and B)P(A \text{ and } B) when calculating P(A or B)P(A \text{ or } B)?
  3. How do we interpret probabilities as fractions in real-world scenarios?
  4. Can probabilities be greater than 1?
  5. What is the significance of mutually exclusive events in probability?

Tip: Always ensure probability values are between 0 and 1.

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory

Formulas

P(A or B) = P(A) + P(B) - P(A and B)

Theorems

Addition Rule for Probability

Suitable Grade Level

Grades 9-12