Question

A 5-column table has 4 rows. The first column has entries A, B, C, Total. The second column is labeled X with entries 10, 110, 60, 180. The third column is labeled Y with entries 80, 44, 59, 183. The fourth column is labeled Z with entries 61, 126, 110, 297. The fifth column is labeled Total with entries 151, 280, 229, 660.
Which statement is true about whether Z and B are independent events?

Z and B are independent events because P(Z∣B) = P(Z).
Z and B are independent events because P(Z∣B) = P(B).
Z and B are not independent events because P(Z∣B) ≠ P(Z).
Z and B are not independent events because P(Z∣B) ≠ P(B).

Answer 1

Z and B are independent events because P(Z|B) = P(Z).

Reason: P(Z|B) = 126/280 = 0.45, and P(Z) = 297/660 = 0.45, so P(Z|B) = P(Z) (while P(B) = 280/660 ≈ 0.424, which is different).