Probability and Playing Cards

Probability and playing cards is an important segment in probability. Here different types of examples will help the students to understand the problems on probability with playing cards.
All the solved questions are pertains to a standard deck of well-shuffled 52 cards playing cards.

Worked-out Examples on Probability and playing cards

1. The king, queen and jack of clubs are removed from a deck of 52 playing cards and then shuffled. A card is drawn from the remaining cards. Find the probability of getting:

(i) a heart

(ii) a queen

(iii) a club

(iv) ‘9’ of red color

Solution:

Total number of card in a deck = 52

Card removed king, queen and jack of clubs

Therefore, remaining cards = 52 - 3=49

Therefore, number of favorable outcomes = 49

(i) a heart

Number of hearts in a deck of 52 cards= 13

Therefore, the probability of getting ‘a heart’

               Number of favorable outcomes
^{P(A) =}     Total number of possible outcome

      = 13/49

(ii) a queen

Number of queen = 3

[Since club’s queen is already removed]

Therefore, the probability of getting ‘a queen t’

               Number of favorable outcomes
^{P(B) =}     Total number of possible outcome

      = 3/49

(iii) a club

Number of clubs in a deck in a deck of 52 cards = 13

According to the question, the king, queen and jack of clubs are removed from a deck of 52 playing cards In this case, total number of clubs = 13 - 3 = 10

Therefore, the probability of getting ‘a club’

               Number of favorable outcomes
^{P(C) =}     Total number of possible outcome

      = 10/49

(iv) ‘9’ of red color

Cards of hearts and diamonds are red cards

The card 9 in each suit, hearts and diamonds = 1

Therefore, total number of ‘9’ of red color = 2

Therefore, the probability of getting ‘9’ of red color

               Number of favorable outcomes
^{P(D) =}     Total number of possible outcome

      = 2/49

2. All kings, jacks, diamonds have been removed from a pack of 52 playing cards and the remaining cards are well shuffled. A card is drawn from the remaining pack. Find the probability that the card drawn is:

(i) a red queen

(ii) a face card

(iii) a black card

(iv) a heart

Solution:

Number of kings in a deck 52 cards = 4

Number of jacks in a deck 52 cards = 4

Number of diamonds in a deck 52 cards = 13

Total number of cards removed = (4 kings + 4 jacks + 11 diamonds) = 19 cards

[Excluding the diamond king and jack there are 11 diamonds]

Total number of cards after removing all kings, jacks, diamonds = 52 - 19 = 33

(i) a red queen

Queen of heart and queen of diamond are two red queens

Queen of diamond is already removed.

So, there is 1 red queen out of 33 cards

Therefore, the probability of getting ‘a red queen’

               Number of favorable outcomes
^{P(A) =}     Total number of possible outcome

      = 1/33

(ii) a face card

Number of face cards after removing all kings, jacks, diamonds = 3

Therefore, the probability of getting ‘a face card’

               Number of favorable outcomes
^{P(B) =}     Total number of possible outcome

      = 3/33
      = 1/11

(iii) a black card

Cards of spades and clubs are black cards.

Number of spades = 13 - 2 = 11, since king and jack are removed

Number of clubs = 13 - 2 = 11, since king and jack are removed

Therefore, in this case, total number of black cards = 11 + 11 = 22

Therefore, the probability of getting ‘a black card’

               Number of favorable outcomes
^{P(C) =}     Total number of possible outcome

      = 22/33
      = 2/3

(iv) a heart

Number of hearts = 13

Therefore, in this case, total number of hearts = 13 - 2 = 11, since king and jack are removed

Therefore, the probability of getting ‘a heart card’

               Number of favorable outcomes
^{P(D) =}     Total number of possible outcome

      = 11/33
      = 1/3

3. A card is drawn from a well-shuffled pack of 52 cards. Find the probability that the card drawn is:

(i) a red face card

(ii) neither a club nor a spade

(iii) neither an ace nor a king of red color

(iv) neither a red card nor a queen

(v) neither a red card nor a black king.

Solution:

Total number of card in a pack of well-shuffled cards = 52

(i) a red face card

Cards of hearts and diamonds are red cards.

Number of face card in hearts = 3

Number of face card in diamonds = 3

Total number of red face card out of 52 cards = 3 + 3 = 6

Therefore, the probability of getting ‘a red face card’

               Number of favorable outcomes
^{P(A) =}     Total number of possible outcome

      = 6/52
      = 3/26

(ii) neither a club nor a spade

Number of clubs = 13

Number of spades = 13

Number of club and spade = 13 + 13 = 26

Number of card which is neither a club nor a spade = 52 - 26 = 26

Therefore, the probability of getting ‘neither a club nor a spade’

               Number of favorable outcomes
^{P(B) =}     Total number of possible outcome

      = 26/52
      = 1/2

(iii) neither an ace nor a king of red color

Number of ace in a deck 52 cards = 4

Number of king of red color in a deck 52 cards = (1 diamond king + 1 heart king) = 2

Number of ace and king of red color = 4 + 2 = 6

Number of card which is neither an ace nor a king of red color = 52 - 6 = 46

Therefore, the probability of getting ‘neither an ace nor a king of red color’

               Number of favorable outcomes
^{P(C) =}     Total number of possible outcome

      = 46/52
      = 23/26

(iv) neither a red card nor a queen

Number of hearts in a deck 52 cards = 13

Number of diamonds in a deck 52 cards = 13

Number of queen in a deck 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

Therefore, the probability of getting ‘neither a red card nor a queen’

               Number of favorable outcomes
^{P(D) =}     Total number of possible outcome

      = 24/52
      = 6/13

(v) neither a red card nor a black king.

Number of hearts in a deck 52 cards = 13

Number of diamonds in a deck 52 cards = 13

Number of black king in a deck 52 cards = (1 king of spade + 1 king of club) = 2

Total number of red card and black king = 13 + 13 + 2 = 28

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

Therefore, the probability of getting ‘neither a red card nor a black king’

               Number of favorable outcomes
^{P(E) =}     Total number of possible outcome

      = 24/52
      = 6/13

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