# Two standard dice are rolled, what is the probability of rolling a pair (both the same number)?

Jun 30, 2018

The probability of rolling the pair

$P \left(E\right) = \frac{1}{6} \approx 0.17$

#### Explanation:

We note that, if a die is rolled the sample space:

$S = \left\{1 , 2 , 3 , 4 , 5 , 6\right\}$

We know that ,if two dice rolled the sample space :

$S = \left\{1 , 2 , 3 , 4 , 5 , 6\right\} \times \left\{1 , 2 , 3 , 4 , 5 , 6\right\}$

:.S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots} \left(2 , 1\right) , \left(2 , 2\right) , \left(2 , 3\right) , \left(2 , 4\right) , \left(2 , 5\right) , \left(2 , 6\right) ,$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots} \left(3 , 1\right) , \left(3 , 2\right) , \left(3 , 3\right) , \left(3 , 4\right) , \left(3 , 5\right) , \left(3 , 6\right) ,$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots} \left(4 , 1\right) , \left(4 , 2\right) , \left(4 , 3\right) , \left(4 , 4\right) , \left(4 , 5\right) , \left(4 , 6\right) ,$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots} \left(5 , 1\right) , \left(5 , 2\right) , \left(5 , 3\right) , \left(5 , 4\right) , \left(5 , 5\right) , \left(5 , 6\right) ,$
color(white)(...............)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

i.e. When two dice are tossed together ,

the possible pairs are : color(blue)(n=6xx6=36

Let ,event $E =$the same number on both dice.

$\implies E = \left\{\begin{matrix}1 & 1 \\ 2 & 2 \\ 3 & 3 \\ 4 & 4 \\ 5 & 5 \\ 6 & 6\end{matrix}\right\}$

$\implies$no. of comeout pair color(blue)( r=6

So , the probability of rolling the pair $= P \left(E\right) = \frac{r}{n}$

$\implies P \left(E\right) = \frac{6}{36}$

$\implies P \left(E\right) = \frac{1}{6} \approx 0.17$

Jun 30, 2018

$\frac{1}{6}$

#### Explanation:

Since we are rolling two six-sided dice, there are $6 \times 6$, or $36$ different possibilities. This will be our denominator.

How many ways can we get double numbers?

$\left\{1 , 1\right\} , \left\{2 , 2\right\} , \left\{3 , 3\right\} , \left\{4 , 4\right\} , \left\{5 , 5\right\} , \left\{6 , 6\right\}$

We have $6$ different ways of getting double numbers. This is our numerator.

Thus, the probability of rolling pairs is $\frac{6}{36}$ or

$\frac{1}{6}$

Hope this helps!