(i) a black king Total numbers of cards are 52 

Number of black king cards = 2 

Probability of getting black king cards is = \(\frac{Total\,number\,of\,blackking\,cards}{Total\,number\,of\,cards}\) = \(\frac{2}{52}\) = \(\frac{1}{26}\)

Therefore Probability of getting black king cards is = \(\frac{1}{26}\) 

(ii) either a black card or a king 

Total numbers of cards are 52 

Number of either a black card or a king = 28 

Probability of getting either a black card or a king is = \(\frac{Total\,number\,of\,blackking\,cards}{Total\,number\,of\,cards}\) =\(\frac{28}{52}\) = \(\frac{7}{13}\)

Therefore Probability of getting either a black card or a king is = \(\frac{7}{13}\)

(iii) black and a king 

Total numbers of cards are 52 

Number of black and a king = 2 

Probability of getting black and a king is = \(\frac{Total\,number\,of\,blackking\,cards}{Total\,number\,of\,cards}\) = \(\frac{2}{52}\) = \(\frac{1}{26}\) 

Therefore Probability of getting black and a king is = \(\frac{1}{26}\)

(iv) a jack, queen or a king 

Total numbers of cards are 52 

Number of a jack, queen or a king = 12 

Probability of getting a jack, queen or a king is = \(\frac{Total\,number\,of\,blackking\,cards}{Total\,number\,of\,cards}\) = \(\frac{12}{52}\) = \(\frac{3}{13}\)

Therefore Probability of getting a jack, queen or a king is = \(\frac{3}{13}\)

(v) neither a heart nor a king 

Total numbers of cards are 52 

Total number of heart cards = 13

Probability of getting a heart is = \(\frac{Total\,number\,of\,blackking\,cards}{Total\,number\,of\,cards}\) = \(\frac{4}{52}\) = \(\frac{1}{13}\)

Total probability of getting a heart and a king = \(\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}\)

Therefore probability of getting neither a heart nor a king = \(1-\frac{4}{13}=\frac{9}{13}\)