Consider the following events:
X = The total score is 8
Y = one of
the dice has thrown 4.
We should find the probability P(X | Y).
P(X | Y) = P(XY) / P(Y)
Let us compute P(Y).
A be the value of the first dice, and B be the value of second dice.
there are 36 possible pairs (A,B).
Among them there are 10 pairs
(1,4), (2,4), (3,4), (4,4), (5,4), (6,4),
(4,3), (4,5), (4,6),
Each pair has teh same probability, therefore
P(Y) = 10/36.
Now compute P(XY).
XY = The
total score is 8 and one of the dice has thrown 4.
This implies that
another dice has thrown 8-4=4, and so XY corresponds to a unique pair
P(XY) = 1/36.
Thus we obtain that
| Y) = P(XY) / P(Y) = (1/36) / (10/36) = 1/10 = 0.1.