Two dice are thrown at the same time. Find the probability of getting:
$(i)$ same number on both dice.
$(ii)$ different numbers on both dice.

(A) When two dice are thrown at the same time,the total number of possible outcomes is $6 \times 6 = 36$.
$(i)$ Let $E_1$ be the event of getting the same number on both dice.
The favorable outcomes are $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)$.
Number of favorable outcomes $= 6$.
Probability $P(E_1) = \frac{6}{36} = \frac{1}{6}$.
$(ii)$ Let $E_2$ be the event of getting different numbers on both dice.
This is the complement of event $E_1$.
Number of favorable outcomes $= 36 - 6 = 30$.
Probability $P(E_2) = \frac{30}{36} = \frac{5}{6}$.