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Question
A pair of dice is thrown. If sum of the numbers is an even number, what is the probability that it is a perfect square?
Solution
When two dice are thrown simultaneously, 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 A be the event that sum of the numbers is an even number.
∴ A = {(1, 1), (1, 3), (1, 5) (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}
∴ n(A) = 18
∴ P(A) = `("n"("A"))/("n"("S")) = 18/36`
Let B be the event that sum of outcomes is a perfect square
B = {(1, 3), (2, 2), (3, 1), (3, 6), (4, 5), (5, 4), (6, 3)}
∴ n(B) = 7
∴ P(B) = `("n"("B"))/("n"("S")) = 7/36`
∴ A ∩ B is the event that the sum of the numbers is an even perfect square.
∴ A ∩ B = {(1, 3), (2, 2), (3, 1)}
∴ n(A ∩ B) = 3
∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 3/36`
Now, Probability that the sum of the numbers is a Perfect square given that it is even is given by,
`"P"("B"/"A") = ("P"("A" ∩ "B"))/("P"("A")`
= `(3/36)/(18/36)`
= `3/18`
= `1/6`
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