19 A fair die is rolled three times What is the probability

19. A fair die is rolled three times. What is the probability that it shows an even number in the first toss, an odd number in the second toss, and a 1 on the third toss? Assume that the outcomes of the tosses are independent.

Solution

Below is the sample space of all 216 outcomes

(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
(1,2,1) (1,2,2) (1,2,3) (1,2,4) (1,2,5) (1,2,6)
(1,3,1) (1,3,2) (1,3,3) (1,3,4) (1,3,5) (1,3,6)
(1,4,1) (1,4,2) (1,4,3) (1,4,4) (1,4,5) (1,4,6)
(1,5,1) (1,5,2) (1,5,3) (1,5,4) (1,5,5) (1,5,6)
(1,6,1) (1,6,2) (1,6,3) (1,6,4) (1,6,5) (1,6,6)
(2,1,1) (2,1,2) (2,1,3) (2,1,4) (2,1,5) (2,1,6)
(2,2,1) (2,2,2) (2,2,3) (2,2,4) (2,2,5) (2,2,6)
(2,3,1) (2,3,2) (2,3,3) (2,3,4) (2,3,5) (2,3,6)
(2,4,1) (2,4,2) (2,4,3) (2,4,4) (2,4,5) (2,4,6)
(2,5,1) (2,5,2) (2,5,3) (2,5,4) (2,5,5) (2,5,6)
(2,6,1) (2,6,2) (2,6,3) (2,6,4) (2,6,5) (2,6,6)
(3,1,1) (3,1,2) (3,1,3) (3,1,4) (3,1,5) (3,1,6)
(3,2,1) (3,2,2) (3,2,3) (3,2,4) (3,2,5) (3,2,6)
(3,3,1) (3,3,2) (3,3,3) (3,3,4) (3,3,5) (3,3,6)
(3,4,1) (3,4,2) (3,4,3) (3,4,4) (3,4,5) (3,4,6)
(3,5,1) (3,5,2) (3,5,3) (3,5,4) (3,5,5) (3,5,6)
(3,6,1) (3,6,2) (3,6,3) (3,6,4) (3,6,5) (3,6,6)
(4,1,1) (4,1,2) (4,1,3) (4,1,4) (4,1,5) (4,1,6)
(4,2,1) (4,2,2) (4,2,3) (4,2,4) (4,2,5) (4,2,6)
(4,3,1) (4,3,2) (4,3,3) (4,3,4) (4,3,5) (4,3,6)
(4,4,1) (4,4,2) (4,4,3) (4,4,4) (4,4,5) (4,4,6)
(4,5,1) (4,5,2) (4,5,3) (4,5,4) (4,5,5) (4,5,6)
(4,6,1) (4,6,2) (4,6,3) (4,6,4) (4,6,5) (4,6,6)
(5,1,1) (5,1,2) (5,1,3) (5,1,4) (5,1,5) (5,1,6)
(5,2,1) (5,2,2) (5,2,3) (5,2,4) (5,2,5) (5,2,6)
(5,3,1) (5,3,2) (5,3,3) (5,3,4) (5,3,5) (5,3,6)
(5,4,1) (5,4,2) (5,4,3) (5,4,4) (5,4,5) (5,4,6)
(5,5,1) (5,5,2) (5,5,3) (5,5,4) (5,5,5) (5,5,6)
(5,6,1) (5,6,2) (5,6,3) (5,6,4) (5,6,5) (5,6,6)
(6,1,1) (6,1,2) (6,1,3) (6,1,4) (6,1,5) (6,1,6)
(6,2,1) (6,2,2) (6,2,3) (6,2,4) (6,2,5) (6,2,6)
(6,3,1) (6,3,2) (6,3,3) (6,3,4) (6,3,5) (6,3,6)
(6,4,1) (6,4,2) (6,4,3) (6,4,4) (6,4,5) (6,4,6)
(6,5,1) (6,5,2) (6,5,3) (6,5,4) (6,5,5) (6,5,6)
(6,6,1) (6,6,2) (6,6,3) (6,6,4) (6,6,5) (6,6,6)

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Number of even number in the first toss, an odd number in the second toss, and a 1 on the third toss =
{
(2,1,1), (2,3,1),(2,5,1),(4,1,1),(4,3,1),(4,5,1),(6,1,1),(6,3,1),(6,5,1)
}
P( even in first toss, an odd number in the second toss, and a 1 on the third toss ) = 6/216 = 1/ 36 = 0.0277