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A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:

(i) a black king

(ii) either a black card or a king

(iii) black and a king

(iv) a jack, queen or a king

(v) neither a heart nor a king

(vi) spade or an ace

(vii) neither an ace nor a king

(viii) neither a red card nor a queen

(ix) other than an ace

(x) a ten

(xi) a spade

(xii) a black card

(xiii) the seven of clubs

(xiv) jack

(xv) the ace of spades

(xvi) a queen

(xvii) a heart

(xviii)a red card

Answer:

(i) A black king

Total number of cards are 52 Number of black king cards = 2

Probability of getting black king cards is = Total number of black king cards/Total number of cards

= 2/52

= 1/26

∴ Probability of getting black king cards is 1/26

(ii) Either a black card or a king Total number 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 = Total number of either black or king card/Total number of cards

= 28/52

= 7/13

∴ Probability of getting either a black card or a king is 7/13

(iii) Black and a king

Total number of cards are 52 Number of black and a king = 2

Probability of getting black and a king is = Total number of black and king card/Total number of cards

= 2/52

= 1/26

∴ Probability of getting black and a king is 1/26

(iv) a jack, queen or a king Total number of cards are 52

Number of a jack, queen or a king = 12

Probability of getting a jack, queen or a king is = Total number of jack, queen or king card/Total number of cards

= 12/52

= 3/13

∴ Probability of getting a jack, queen or a king is 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 = Total number of hearts/Total number of cards

= 13/52

= 1/4

Total number of king cards = 4

Probability of getting a king is = Total number of king card/Total number of cards

= 4/52

= 1/13

Total probability of getting a heart and a king = 13/52 + 4/52 – 1/52

= (13+4-1)/52

= 16/52

= 4/13

∴ Probability of getting neither a heart nor a king = 1 – 4/13 = (13-4)/13 = 9/13

(vi) Spade or an ace

Total numbers of cards are 52 Number of spade cards = 13

Probability of getting spade cards is = Total number of spade card/Total number of cards

= 13/52

Number of ace cards = 4

Probability of getting ace cards is = Total number of ace card/Total number of cards

= 4/52

= 1/13

Probability of getting ace and spade cards is = Total number of ace and spade card/Total number of cards

= 1/52

Probability of getting an ace or spade cards is = 13/52 + 4/52 – 1/52

= (13+4-1)/52

= 16/52

= 4/13

∴ Probability of getting an ace or spade cards is = 4/13

(vii) Neither an ace nor a king Total numbers of cards are 52 Number of king cards = 4

Number of ace cards = 4

Total number of cards = 4 + 4 = 8

Total number of neither an ace nor a king are = 52 – 8 = 44

Probability of getting neither an ace nor a king is = Total number of neither ace nor king card/Total number of cards

= 44/52

= 11/13

∴ Probability of getting neither an ace nor a king is 11/13

(viii) Neither a red card nor a queen Total numbers of cards are 52

Red cards include hearts and diamonds Number of hearts in a deck of 52 cards = 13 Number of diamonds in a deck of 52 cards = 13 Number of queen in a deck of 52 cards = 4

Total number of red card and queen = 13 + 13 + 2 = 28 [since queen of heart and queen of diamond are removed]

Number of card which is neither a red card nor a queen = 52 - 28 = 24

Probability of getting neither a king nor a queen is = Total number of neither red nor queen card/Total number of cards

= 24/52

= 6/13

∴ Probability of getting neither a king nor a queen is 6/13

(ix) Other than an ace

Total numbers of cards are 52 Total number of ace cards = 4

Total number of non-ace cards = 52-4 = 48

Probability of getting non-ace is = Total number of non-ace cards/Total number of cards

= 48/52

= 12/13

∴ Probability of getting non-ace card is 12/13

(x) A ten

Total numbers of cards are 52 Total number of ten cards = 4

Probability of getting ten cards is = Total number of ten cards/Total number of cards

= 4/52

= 1/13

∴ Probability of getting ten card is 1/13

(xi) A spade

Total numbers of cards are 52 Total number of spade cards = 13

Probability of getting spade is = Total number of spade cards/Total number of cards

= 13/52

= 1/4

∴ Probability of getting a spade is 1/4

(xii) A black card

Total numbers of cards are 52

Cards of spades and clubs are black cards. Number of spades = 13

Number of clubs = 13

Total number of black card out of 52 cards = 13 + 13 = 26

Probability of getting black cards is = Total number of black cards/Total number of cards

= 26/52

= 1/2

∴ Probability of getting a black card is 1/2

(xiii) The seven of clubs Total numbers of cards are 52

Total number of the seven of clubs cards = 1

Probability of getting the seven of clubs cards is = Total number of the seven of club cards/ Total numbers of cards

= 1/52

∴ Probability of the seven of club card is 1/52

(xiv) Jack

Total numbers of cards are 52 Total number of jack cards = 4

Probability of getting jack cards is = Total number of jack cards/ Total numbers of cards

= 4/52

= 1/13

∴ Probability of the jack card is 1/13

(xv) The ace of spades

Total numbers of cards are 52

Total number of the ace of spades cards = 1

Probability of getting ace of spade cards is = Total number of ace of spade cards/ Total numbers of cards

= 1/52

∴ Probability of the ace of spade card is 1/52

(xvi) A queen

Total numbers of cards are 52 Total number of queen cards = 4

Probability of getting queen cards is = Total number of queen cards/Total numbers of cards

= 4/52

= 1/13

∴ Probability of a queen card is 1/13

(xvii) A heart

Total numbers of cards are 52 Total number of heart cards = 13

Probability of getting queen cards is = Total number of heart cards/Total numbers of cards

= 13/52

= 1/4

∴ Probability of a heart card is 1/4

(xviii) A red card

Total numbers of cards are 52

Total number of red cards = 13+13 = 26

Probability of getting queen cards is = Total number of red cards/Total numbers of cards

= 26/52

= 1/2 ∴ Probability of a red card is 1/2

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