roll a single die 3 times What is the probability that at le

roll a single die 3 times. What is the probability that at least two of the rolls had the same value?

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 atleast 2 of the rows has same value = { (1,2,1)   (1,2,2)   (1,3,3)   (1,4,4)   (1,5,5)   (1,6,6)
(1,3,1)   (2,1,2)   (2,2,3)   (2,2,4)   (2,2,5)   (2,2,6)
(1,4,1)   (2,2,2)   (2,3,3)   (2,4,4)   (2,5,5)   (2,6,6)
(1,5,1)   (2,3,2)   (3,1,3)   (3,3,4)   (3,3,5)   (3,3,6)
(1,6,1)   (2,4,2)   (3,2,3)   (3,4,4)   (3,5,5)   (3,6,6)
(2,1,1)   (2,5,2)   (3,3,3)   (4,1,4)   (4,4,5)   (4,4,6)
(2,2,1)   (2,6,2)   (3,4,3)   (4,2,4)   (4,5,5)   (4,6,6)
(3,1,1)   (3,2,2)   (3,5,3)   (4,3,4)   (5,1,5)   (5,5,6)
(3,3,1)   (3,3,2)   (3,6,3)   (4,4,4)   (5,2,5)   (5,6,6)
(4,1,1)   (4,2,2)   (4,3,3)   (4,5,4)   (5,3,5)   (6,1,6)
(4,4,1)   (4,4,2)   (4,4,3)   (4,6,4)   (5,4,5)   (6,2,6)
(5,1,1)   (5,2,2)   (5,3,3)   (5,4,4)   (5,5,5)   (6,3,6)
(5,5,1)   (5,5,2)   (5,5,3)   (5,5,4)   (5,6,5)   (6,4,6)
(6,1,1)   (6,2,2)   (6,3,3)   (6,4,4)   (6,5,5)   (6,5,6)
(6,6,1)   (6,6,2)   (6,6,3)   (6,6,4)   (6,6,5)   (6,6,6) }

P( Probability of atleast 2 of the rows has same value ) = No.of such occurence/ total sample space= 90/216 = 0.4166