Question

If you roll two dice, what is the probability of rolling a $3$ and an even number?

Answer 1

Step-1: Find Sample space:

The sample space for rolling a dice is,

$$\begin{gathered} \mathrm{S}=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right), \\ \left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right), \\ \left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right), \\ \left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right), \\ \left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right), \\ \left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\right\} \end{gathered}$$

$\Rightarrow \mathrm{n}\left(\mathrm{S}\right)=36$

Step-2: Find the probability of $\mathbf{3}$ and an even number:

Consider A to be the probability of getting $3$ and an even number.

$\Rightarrow \mathrm{A}=\left\{\left(3,2\right),\left(3,4\right),\left(3,6\right)\right\}$

$\Rightarrow \mathrm{n}\left(\mathrm{A}\right)=3$

The probability of the event $\mathrm{A}$ is

$$\begin{gathered} \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{\mathrm{n}\left(\mathrm{A}\right)}{\mathrm{n}\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{3}{36} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{1}{12} \end{gathered}$$

The probability of getting $3$ and an even number is same as the probability of getting an even number and $3$.

The probability of getting $3$ and an even number in any order is,

$$\begin{gathered} \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{1}{12}+\frac{1}{12} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{2}{12} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{1}{6} \end{gathered}$$

Hence, the probability of getting $\mathbf{3}$ and an even number on rolling two dice is $\frac{\mathbf{1}}{\mathbf{6}}$.