Three dices are rolled 1what is the probability that the su

Three dices are rolled ,

1.what is the probability that the sum of them are more than 12

2. what is the probability that the sum of them are even

3. what is the probability that the product of them are even

4. if known the sum of three of them are even, then what is the probability that one of them is odd

Solution

Below is the sample space of all 216 outcomes, and the 56 red
ones are the successful outcomes that gives the sum more than 12

(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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Probability of sum more than 12 = P ( SUM > 12 ) = 56/216 = 0.25925