Question

Use the Venn diagram to calculate conditional probabilities.



Circles D, E, and F overlap. Circle D contains 13, circle E contains 4, and circle F contains F. The overlap of D and E contains 6, the overlap of E and F contains 7, and the overlap of F and D contains 5. The overlap of all 3 circles contains 5. Number 3 is outside of the circles.

Which conditional probabilities are correct? Check all that apply.

P(D | F) = StartFraction 6 Over 34 EndFraction
P(E | D) = StartFraction 7 Over 25 EndFraction
P(D | E) = StartFraction 7 Over 25 EndFraction
P(F | E) = StartFraction 8 Over 18 EndFraction
P(E | F) = StartFraction 13 Over 21 EndFraction

Answer 1

None are correct.

Reason: From the diagram the pairwise intersections (including the triple overlap) are
- D∩E = 6
- E∩F = 7
- D∩F = 5
and the circle totals are D = 13, E = 14, F = 12 (so the exclusive pieces work out as D only = 7, E only = 6, F only = 5). Therefore the correct conditional probabilities are

- P(D | F) = |D∩F| / |F| = 5/12
- P(E | D) = |E∩D| / |D| = 6/13
- P(D | E) = |D∩E| / |E| = 6/14 = 3/7
- P(F | E) = |F∩E| / |E| = 7/14 = 1/2
- P(E | F) = |E∩F| / |F| = 7/12

None of the given fractions match these values.