Question

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

Answer 1

Interpreting the diagram as region counts:
A only = 12, B only = 11, C only = 4, A∩B only = 5, B∩C only = 3, A∩C only = 6, A∩B∩C = 8. Total = 49.

- P(A|C) = P(A∩C)/P(C) = (6+8)/(4+6+3+8) = 14/21 = 2/3. (Correct)
- P(C) = (4+6+3+8)/49 = 21/49 = 3/7. (Correct)

The other three options are incorrect.