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Question
Two dice are thrown together. Let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent?
Solution
When two dice are thrown, the sample space is
S = {(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),
(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 (S) = 36
Let event A: Getting 6 on the first die.
∴ A = {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ n(A) = 6
∴ P(A) = `("n"("A"))/("n"("S")) = 6/36 = 1/6`
Let event B: Gettting 2 on the second die.
∴ B = {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}
∴ n(B) = 6
∴ P(B) = `("n"("B"))/("n"("S")) = 6/36 = 1/6`
Now, A ∩ B = {(6, 2)}
∴ n(A ∩ B) = 1
∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 1/36` ...(i)
P(A) × P(B) = `1/6 xx 1/6 = 1/36` ...(ii)
From (i) and (ii), we ge
P(A ∩ B) = P(A) × P(B)
∴ A and B are independent events.
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