A black and a red die are rolled together.
⇒ S = [(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)]
We need to find the probability of obtaining the sum 8, given that red die resulted in a number less than 4.
Let A be the event that a number on red die is less than 4.
Let B be the event that the sum of the numbers is 8.
A = [(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)] and B = [(2,6) , (3,5) , (4,4) , (5,3) , (6,2)]
⇒ A ∩ B = {(2,6) , (3,5)}
n (A) = 18, n (B) = 5, n (A ∩ B ) = 2, and n(S) = 36
P (A) =

18

36

, P (B) =

5

36

and P (A ∩ B) =

2

36

The conditional probability of obtaining the sum equal to 8, given that the red die resulted in a number less than 4 is given by,
⇒ P (B|A) =

P(A∩B)

P(A)

=

2 36

18 36

=

1

9