Given, two events A and B are independent such that A = {(x, y) : x + y = 11} B = {(x, y) : x ≠ 5}
Now,
A = {(5, 6), (6, 5)}
B = {(1, 1), (1, 2), (1, 3), (1, 5), (1, 6), (2, 1), (2, 2), (2, 4), (2, 5), (2, 6), (3, 1), (3, 3), (3, 4), (3, 5), (3, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
n(A) = 2, n(B) = 30 and n(A ⋂ B) = 1
So, P(A) = 2/36 = 1/18 and P(B) = 30/36 = 5/6
Now,
P(A).P(B) = 1/18. 5/6 = 5/108 and P(A ⋂ B) 1/36
As P(A). P(B) ≠ P(A ⋂ B)
Therefore, events A and B are not independent.
