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Question
Refer to Question 1 above. If the die were fair, determine whether or not the events A and B are independent.
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Solution
We have A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
∴ n(A) = 6 and n(S) = 6 × 6 = 36
So, P(A) = `("n"("A"))/("n"("S")) = 6/36 = 1/6`
And B = {(4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6)}
n(B) = 6 and n(S) = 36
∴ P(B) = `("n"("B"))/("n"("S")) = 6/36 = 1/6`
A ∩ B = {(5, 5), (6, 6)}
∴ P(A ∩ B) = `2/36 = 1/18`
Therefore, if A and B are independent
Then P(A ∩ B) = P(A) . P(B)
⇒ `1/18 ≠ 1/6 xx 1/6`
⇒ `1/18 ≠ 1/36`
Hence, A and B are not independent events.
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