Answer to Question #9140 in Statistics and Probability for Akshay

Question

Two dice are thrown. What is the conditional probability of a total score of at least 8, given that one of the dice has thrown 4 .

Answer 1

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).

By
definition

P(X | Y) = P(XY) / P(Y)


Let us compute P(Y).
Let
A be the value of the first dice, and B be the value of second dice.
Then
there are 36 possible pairs (A,B).

Among them there are 10 pairs
containing 4:
(1,4), (2,4), (3,4), (4,4), (5,4), (6,4),
(4,1), (4,2),
(4,3), (4,5), (4,6),

Each pair has teh same probability, therefore

P(Y) = 10/36.


Now compute P(XY).
Notice that

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
(4,4).
Hence
P(XY) = 1/36.


Thus we obtain that

P(X
| Y) = P(XY) / P(Y) = (1/36) / (10/36) = 1/10 = 0.1.