$A$ die is thrown once. Find the probability of getting:
$(i)$ a prime number;
$(ii)$ a number lying between $2$ and $6$;
$(iii)$ an odd number.

(N/A) The set of all possible outcomes when a fair die is thrown is $S = \{1, 2, 3, 4, 5, 6\}$.
The total number of possible outcomes is $n(S) = 6$.
$(i)$ The prime numbers on a die are $\{2, 3, 5\}$.
The number of favorable outcomes is $3$.
Probability $P(\text{prime}) = \frac{3}{6} = \frac{1}{2}$.
$(ii)$ The numbers lying between $2$ and $6$ are $\{3, 4, 5\}$.
The number of favorable outcomes is $3$.
Probability $P(\text{between } 2 \text{ and } 6) = \frac{3}{6} = \frac{1}{2}$.
$(iii)$ The odd numbers on a die are $\{1, 3, 5\}$.
The number of favorable outcomes is $3$.
Probability $P(\text{odd}) = \frac{3}{6} = \frac{1}{2}$.