Question

Two dice are tossed. Find whether the following two events A and B are independent: A = {(x, y): x + y = 11} B = {(x, y): x ≠ 5} where (x, y) denotes a typical sample point.

Answer 1

Given that, A = {(x, y): x + y = 11} B = {(x, y): x ≠ 5}

∴ A = {(5, 6), (6, 5)},

B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

⇒ n(A) = 2

n(B) = 30

And n(A ∩ B) = 1

∴ P(A) = `2/36 = 1/18` and P(B) = `30/36 = 5/6`

⇒ P(A) . P(B) = `1/18 * 5/6 = 5/108`

And P(A ∩ B) = `1/36`

Since P(A) . P(B) ≠ P(A ∩ B) 

Hence, A and B are not independent.