Question

Let Y be the random variable denoting the sum of the points on the upturned faces of the dice when a pair of dice is rolled. Find the probability that the sum of points of the upturned faces is 5

Answer 1

The sample space $S$ is given below.

$S=\begin{Bmatrix} (1,1) & (2,1)&(3,1)&(4,1)&(5,1)&(6,1) \\ (1,2) & (2,2)&(3,2)&(4,2)&(5,2)&(6,2)\\ (1,3)&(2,3)&(3,3)&(4,3)&(5,3)&(6,3)\\ (1,4)&(2,4)&(3,4)&(4,4)&(5,4)&(6,4)\\ (1,5)&(2,5)&(3,5)&(4,5)&(5,5)&(6,5)\\ (1,6)&(2,6)&(3,6)&(4,6)&(5,6)&(6,6) \end{Bmatrix}$

From the sample space, the points that sum up to 5 are, $\{1,4\},\{2,3\},\{3,2\},\{4,1\}$.

So, out of $n=36$ outcomes, $y=4$ outcomes sum up to 5. The probability, $p(Y=5)$, that the sum of points of the upturned faces is 5 is ${y\over n}={4\over36}={1\over9}$.

Therefore, $p(Y=5)={1\over9}$