Math Problem Statement

Provide an appropriate response.

If P(A) = 0.84, P(B) =0.76 and

P(A or B) = 0.90,

then P(A and B) is:

Question 21 options:

0.83

0.70

0.06

0.14

Solution

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

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)

We are given:

  • P(A)=0.84P(A) = 0.84
  • P(B)=0.76P(B) = 0.76
  • P(A or B)=0.90P(A \text{ or } B) = 0.90

Substitute the known values into the formula:

0.90=0.84+0.76P(A and B)0.90 = 0.84 + 0.76 - P(A \text{ and } B)

Solving for P(A and B)P(A \text{ and } B):

0.90=1.60P(A and B)0.90 = 1.60 - P(A \text{ and } B)

P(A and B)=1.600.90P(A \text{ and } B) = 1.60 - 0.90

P(A and B)=0.70P(A \text{ and } B) = 0.70

Thus, the correct answer is 0.70.

Would you like any further details or clarifications?


Related Questions:

  1. What is the difference between P(A or B)P(A \text{ or } B) and P(A and B)P(A \text{ and } B)?
  2. How can we apply this formula in real-life scenarios such as survey data?
  3. How do we calculate the probability of independent events occurring together?
  4. What happens to P(A and B)P(A \text{ and } B) if P(A or B)P(A \text{ or } B) exceeds P(A)P(A) or P(B)P(B)?
  5. Can this approach be extended to three or more events, and how?

Tip: Remember to always check if events are independent or mutually exclusive, as that affects how probabilities are calculated!

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory
Union and Intersection of Events

Formulas

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

Theorems

Addition rule of probability

Suitable Grade Level

Grades 9-11