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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.
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Solution
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.
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