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Playing cards probability problems based on a well-shuffled deck of 52 cards.

Nội dung chính

    What is the probability of getting diamond in a deck of cards?How many diamonds are in a deck of cards of 52?How many ways can you make a 2 card hand from the diamonds?How many possible 2 cards are in a deck?

Basic concept on drawing a card:

In a pack or deck of 52 playing cards, they are divided into 4 suits of 13 cards each i.e. spades ♠ hearts ♥, diamonds ♦, clubs ♣.

Cards of Spades and clubs are black cards.

Cards of hearts and diamonds are red cards.

The
card in each suit, are ace, king, queen, jack or knaves, 10, 9, 8, 7, 6, 5, 4, 3 and 2.

King, Queen and Jack (or Knaves) are face cards. So, there are 12 face cards in the deck of 52 playing cards.

Worked-out problems on Playing cards probability:

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

(i) ‘2’ of spades

(ii) a jack

(iii) a king of red colour

(iv) a card of
diamond

(v) a king or a queen

(vi) a non-face card

(vii) a black face card

(viii) a black card

(ix) a non-ace

(x) non-face card of black colour

(xi) neither a spade nor a jack

(xii) neither a heart nor a red king

Solution:

In a playing card there are 52 cards.

Therefore the total number of possible outcomes = 52

(i) ‘2’ of spades:

Number of favourable outcomes i.e. ‘2’ of spades is 1 out of 52 cards.

Therefore, probability of getting ‘2’ of spade

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

      = 1/52

(ii) a jack

Number of favourable outcomes i.e. ‘a jack’ is 4 out of 52 cards.

Therefore, probability of getting ‘a jack’

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

      = 4/52
      = 1/13

(iii) a king of red colour

Number of favourable outcomes i.e. ‘a king of red colour’ is 2 out of 52 cards.

Therefore, probability of getting ‘a king of red colour’

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

      = 2/52
      = 1/26

(iv) a card of diamond

Number of favourable outcomes i.e. ‘a card of diamond’ is 13 out of 52 cards.

Therefore, probability of getting ‘a card of diamond’

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

 
    = 13/52
      = 1/4

(v) a king or a queen

Total number of king is 4 out of 52 cards.

Total number of queen is 4 out of 52 cards

Number of favourable outcomes i.e. ‘a king or a queen’ is 4 + 4 = 8 out of 52 cards.

Therefore, probability of getting ‘a king or a queen’

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

      = 8/52
      = 2/13

(vi) a non-face card

Total number of face card out of 52 cards = 3 times 4 = 12

Total number of non-face card out of 52 cards = 52 – 12 = 40

Therefore, probability of getting ‘a non-face card’

               Number of favorable outcomes
P(F) =     Total number of possible outcome

      = 40/52
      = 10/13

(vii) a black face card:

Cards of Spades and Clubs are black cards.

Number of face card in spades (king, queen and jack or knaves) = 3

Number of face card in clubs (king, queen and jack or knaves) = 3

Therefore, total number of black face card out of 52 cards = 3 + 3 = 6

Therefore, probability of getting ‘a black face card’

         
     Number of favorable outcomes
P(G) =     Total number of possible outcome

      = 6/52
      = 3/26

(viii) a black card:

Cards of spades and clubs are black cards.

Number of spades = 13

Number of clubs = 13

Therefore, total number of black card out of 52 cards = 13 + 13 = 26

Therefore, probability of getting ‘a black card’

 
             Number of favorable outcomes
P(H) =     Total number of possible outcome

      = 26/52
      = 1/2

(ix) a non-ace:

Number of ace cards in each of four suits namely spades, hearts, diamonds and clubs = 1

Therefore, total number of ace cards out of 52 cards = 4

Thus, total number of non-ace cards out of 52 cards = 52 – 4

=
48

Therefore, probability of getting ‘a non-ace’

               Number of favorable outcomes
P(I) =     Total number of possible outcome

      = 48/52
      = 12/13

(x) non-face card of black colour:

Cards of spades and clubs are black cards.

Number of spades = 13

Number of clubs = 13

Therefore, total number of
black card out of 52 cards = 13 + 13 = 26

Number of face cards in each suits namely spades and clubs = 3 + 3 = 6

Therefore, total number of non-face card of black colour out of 52 cards = 26 – 6 = 20

Therefore, probability of getting ‘non-face card of black colour’

               Number of favorable outcomes
P(J) =     Total number of possible outcome

     
= 20/52
      = 5/13

(xi) neither a spade nor a jack

Number of spades = 13

Total number of non-spades out of 52 cards = 52 – 13 = 39

Number of jack out of 52 cards = 4

Number of jack in each of three suits namely hearts, diamonds and clubs = 3

[Since, 1 jack is already included in the 13 spades so, here we will take number of jacks is 3]

Neither a spade nor a jack = 39 – 3 = 36

Therefore, probability of
getting ‘neither a spade nor a jack’

               Number of favorable outcomes
P(K) =     Total number of possible outcome

      = 36/52
      = 9/13

(xii) neither a heart nor a red king

Number of hearts = 13

Total number of non-hearts out of 52 cards = 52 – 13 = 39

Therefore, spades, clubs and diamonds are the 39 cards.

Cards of hearts and diamonds are red cards.

Number of red kings in red cards = 2

Therefore, neither a heart nor a red king = 39 – 1 = 38

[Since, 1 red king is already included in the 13 hearts so, here we will take number of red kings is 1]

Therefore, probability of getting ‘neither a heart nor a red king’

               Number of favorable outcomes
P(L) =     Total number of
possible outcome

      = 38/52
      = 19/26

2. A card is drawn random from a
well-shuffled pack of cards numbered 1 to 20. Find the probability of

(i) getting a number less than 7

(ii) getting a number divisible by 3.

Solution:

(i) Total number of possible outcomes = 20 ( since there are cards numbered 1, 2, 3, …, 20).

Number of favourable outcomes for the sự kiện E

                                = number of cards showing less than 7 = 6 (namely
1, 2, 3, 4, 5, 6).

So, P(E) = (fractextrmNumber of Favourable Outcomes for the Event EtextrmTotal Number of Possible Outcomes)

             = (frac620)

             = (frac310).

(ii) Total number of possible outcomes = 20.

Number of favourable outcomes for the sự kiện F

                           
    = number of cards showing a number divisible by 3 = 6 (namely 3, 6, 9, 12, 15, 18).

So, P(F) = (fractextrmNumber of Favourable Outcomes for the Event FtextrmTotal Number of Possible Outcomes)

             = (frac620)

             = (frac310).

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

(i) a king

(ii) neither a queen nor a jack.

Solution:

Total number of possible outcomes = 52 (As there are 52 different cards).

(i) Number of favourable outcomes for the sự kiện E = number of kings in the pack = 4.

So, by definition, P(E) = (frac452)

                                 = (frac113).

(ii) Number of favourable outcomes for the
sự kiện F

                    = number of cards which are neither a queen nor a jack

                    = 52 – 4 – 4, [Since there are 4 queens and 4 jacks].

                    = 44

Therefore, by definition, P(F) = (frac4452)

                       
                  = (frac1113).

These are the basic problems on probability with playing cards.

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What is the probability of getting diamond in a deck of cards?

There are 13 cards of diamond in a 52-card deck. Therefore, the probability of being dealt a diamond is 1/4.

How many diamonds are in a deck of cards of 52?

A standard 52-card deck comprises 13 ranks in each of the four French suits: clubs (♣), diamonds (♦), hearts (♥) and spades (♠).

How many ways can you make a 2 card hand from the diamonds?

Since there are 13 diamonds and we want 2 of them, there are C(13,2) = 78 ways to get the 2 diamonds. Since we want them both to occur the same time, we use the fundamental counting principle and multiply 286 and 78 together to get 22,308 possible hands.

How many possible 2 cards are in a deck?

So, there are 13*13 = 169 ways to draw 2 cards from a deck of 52. D H D D D C D S As you can see, there are 16 different suit combinations. So we have 169 * 16 = 2,704 ways to pick two cards out of a deck of 52.
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