Two fair dice are thrown, find the probability of the following events that the sum of the scores on their upper faces is (a) less than four (b) Multiple of 5 (c) Perfect square

Since Two dice are thrown,
∴ The Sample Space  =
{(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

(a) Let A = Event of getting the sum of the scores on the upper faces of dice is less than 4

∴ A = {(1 , 1) (1 , 2) (2 , 1))

∴ n(A) = 3

∴ P(A) =n(A) / n(S)

∴ P(A) = 3 / 36

∴ P(A) = 1 / 12

(b) Let B = Event of getting the sum of the scores on the upper faces of dice is multiple of 5.

∴ B = { ( 1 , 4) ( 2 , 3) (3 , 2) ( 4 , 1) (4 , 6) ( 5 , 5) ( 6 , 4 ) }

∴ n(B)  = 7

∴ P(B) = n(B) / n(S)

∴ P(B) = 7/36

(c) Let C =  Event of getting the sum of the scores on the upper faces of dice is a perfect square.

C = {(1 , 3) (2 , 2) (3 , 1) (3 , 6) (4 , 5) ( 5 , 4) ( 6 , 3) }

∴ n(C) = 7

∴ P(C ) = n(C) / n(S)

∴ P(C) = 7 / 36

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