Question

Two throws are made, the first with 3 dice and the second with 2 dice. The faces of each die are marked with the number 1 to 6. What is the probability that the total in first throw is not less than 15 and at the same time the total in the second throw is not less than 8?

Answer 1

When 3 dice are thrown, then the sample space S₁ has 6 × 6 × 6 = 216 sample points.
∴ n(S₁) = 216
Let A be the event that the sum of the numbers is not less than 15.
∴ A = {(3,6,6), (4,5,6), (4,6,5), (4,6,6), (5,4,6), (5,5,5), (5,5,6), (5,6,4), (5,6,5), (5,6,6), (6,3,6), (6,4,5), (6,4,6), (6,5,4), (6,5,5), (6,5,6), (6,6,3), (6,6,4), (6,6,5), (6,6,6)}
∴ n(A) = 20

∴ P(A) = `("n"("A"))/("n"("S"_1)) = 20/216 = 5/54`
When 2 dice are thrown, the sample space S₂ has 6 × 6 = 36 sample points.
∴ n(S₂) = 36
Let B be the event that sum of numbers is not less than 8.
∴ B = {(2, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 3), (5, 4), (5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ n(B) = 15

∴ P(B) = `("n"("B"))/("n"("S"_2)) = 15/36 = 5/12`
A ∩ B = Event that the total in the first throw is not less than 15 and at the same time the total in the second throw is not less than 8
∵ A and B are independent events

∴ P(A ∩ B) = P(A).P(B) = `5/54 xx 5/12 = 25/648`