The king,queen,and jack of clubs are removed from a deck of $52$ playing cards and then well shuffled. Now,one card is drawn at random from the remaining cards. Determine the probability that the card is:
$(i)$ a heart
$(ii)$ a king

(A) Total number of cards in a deck = $52$.
After removing the king,queen,and jack of clubs,the number of remaining cards is $n(S) = 52 - 3 = 49$.
$(i)$ Let $E_1$ be the event of getting a heart.
Since there are $13$ hearts in a deck and none were removed,$n(E_1) = 13$.
Therefore,the probability $P(E_1) = \frac{n(E_1)}{n(S)} = \frac{13}{49}$.
$(ii)$ Let $E_2$ be the event of getting a king.
There are $4$ kings in a deck. Since the king of clubs was removed,the number of remaining kings is $n(E_2) = 4 - 1 = 3$.
Therefore,the probability $P(E_2) = \frac{n(E_2)}{n(S)} = \frac{3}{49}$.