Two dice are rolled simultaneously. The probability that the numbers on them are different is

The random experiment is "Rolling two dice"

Let $S$ be the sample space, $n\left(S\right)=6^2=36$

Let $E$ be the event of getting two different numbers on dice

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

There are 30 favourable cases for the event $E$

Hence, $n\left(E\right)=30$

Therefore, the required probability is $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{30}{36}=\boxed{\frac{5}{6}}$