Question

Use the Venn diagram to calculate probabilities.

Circles A, B, and C overlap. Circle A contains 3, circle B contains 9, and circle C contains 6. The overlap of A and B contains 1, the overlap of B and C contains 4, and the overlap of C and A contains 7. The overlap of the 3 circles contains 6. Number 8 is outside of the circles.

Which probability is correct?

P(A|B) = One-half
P(B|A) = StartFraction 7 Over 20 EndFraction
P(A|C) = StartFraction 6 Over 23 EndFraction
P(C|A) = StartFraction 13 Over 17 EndFraction

Answer 1

First find counts and total. Let regions be:
- A only = 3
- B only = 9
- C only = 6
- A∩B only = 1
- B∩C only = 4
- C∩A only = 7
- A∩B∩C = 6
- Outside = 8

Total N = 3+9+6+1+4+7+6+8 = 44.

Useful intersections:
- A∩B = 1+6 = 7
- A∩C = 7+6 = 13
- A = 3+1+7+6 = 17
- B = 9+1+4+6 = 20
- C = 6+7+4+6 = 23

Compute conditionals:
- P(A|B) = P(A∩B)/P(B) = 7/20 (not 1/2).
- P(B|A) = P(A∩B)/P(A) = 7/17 (not 7/20).
- P(A|C) = 13/23 (not 6/23).
- P(C|A) = 13/17 (this one is correct).

So only P(C|A) = 13/17 is correct as given.