Math Problem Statement

Use the Venn diagram to calculate probabilities.

Circles A, B, and C overlap. Circle A contains 12, circle B contains 11, and circle C contains 4. The overlap of A and B contains 5, the overlap of B and C contains 3, and the overlap of C and A contains 6. The overlap of the 3 circles contains 8.

Which probabilities are correct? Select two options.

P(A|C) = Two-thirds P(C|B) = StartFraction 8 Over 27 EndFraction P(A) = StartFraction 31 Over 59 EndFraction P(C) = Three-sevenths P(B|A) = StartFraction 13 Over 27 EndFraction

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Conditional Probability
Venn Diagrams
Set Theory

Formulas

P(A|B) = P(A ∩ B) / P(B)
P(A) = Number of favorable outcomes / Total number of outcomes
Inclusion-Exclusion Principle: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

Theorems

Conditional Probability Theorem
Inclusion-Exclusion Principle

Suitable Grade Level

Grades 10-12