52 Card Deck

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


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 ♣.
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Cards of Spades and clubs are black cards.
Cards of hearts and diamonds are red cards.
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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 theprobability 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 possibleoutcomes = 52
(i) ‘2’ of spades:
Number of favourable outcomes i.e. ‘2’ ofspades is 1 out of 52 cards.
Therefore, probability of getting ‘2’ ofspade 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 kingof red colour’ is 2 out of 52 cards.
Therefore, probability of getting ‘a kingof red colour’ Number of favorable outcomes
P(C) = Total number of possible outcome
= 2/52
= 1/26
52 Card Deck Generator
(iv) a card of diamond
Number of favourable outcomes i.e. ‘a cardof diamond’ is 13 out of 52 cards.
Therefore, probability of getting ‘a cardof 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 kingor a queen’ is 4 + 4 = 8 out of 52 cards.
Therefore, probability of getting ‘a kingor 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 52cards = 52 - 12 = 40
Therefore, probability of getting ‘anon-face card’ Number of favorable outcomes
P(F) = Total number of possible outcome
= 40/52
= 10/13

(vii) a black face card:
Cardsof Spades and Clubs are black cards.
Number of face card in spades (king, queenand jack or knaves) = 3
Number of face card in clubs (king, queen andjack or knaves) = 3
Therefore, total number of black face cardout of 52 cards = 3 + 3 = 6
Therefore, probability of getting ‘a blackface 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 outof 52 cards = 13 + 13 = 26
Therefore, probability of getting ‘a blackcard’ 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 namelyspades, hearts, diamonds and clubs = 1
Therefore, total number of ace cards out of52 cards = 4
Thus, total number of non-ace cards out of52 cards = 52 - 4
= 48
Therefore, probability of getting ‘anon-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 outof 52 cards = 13 + 13 = 26
Number of face cards in each suits namelyspades and clubs = 3 + 3 = 6
Therefore, total number of non-face card ofblack colour out of 52 cards = 26 - 6 = 20
Therefore, probability of getting ‘non-facecard 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 suitsnamely hearts,diamonds and clubs = 3
[Since, 1 jack is already included in the13 spades so, here we will take number of jacks is 3]
Neither a spade nor a jack = 39 - 3 = 36
Therefore, probability of getting ‘neithera 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 arethe 39 cards.
Cardsof 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 inthe 13 hearts so, here we will take number of red kings is 1]
Therefore, probability of getting ‘neithera heart nor a red king’ Number of favorable outcomes
P(L) = Total number of possible outcome
= 38/52
= 19/26
52 Card Deck Suits


2. A card is drawn at random from a well-shuffled pack of cards numbered 1 to 20. Find the probability of
52 Card Deck
(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). Short deck poker book.
Number of favourable outcomes for the event E
= number of cards showing less than 7 = 6 (namely 1, 2, 3, 4, 5, 6).
So, P(E) = (frac{textrm{Number of Favourable Outcomes for the Event E}}{textrm{Total Number of Possible Outcomes}})
= (frac{6}{20})
= (frac{3}{10}).

(ii) Total number of possible outcomes = 20.
Number of favourable outcomes for the event F
= number of cards showing a number divisible by 3 = 6 (namely 3, 6, 9, 12, 15, 18).
So, P(F) = (frac{textrm{Number of Favourable Outcomes for the Event F}}{textrm{Total Number of Possible Outcomes}})
= (frac{6}{20})
= (frac{3}{10}).

3. A card is drawn at 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 event E = number of kings in the pack = 4.
So, by definition, P(E) = (frac{4}{52})
= (frac{1}{13}).

(ii) Number of favourable outcomes for the event 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) = (frac{44}{52})
= (frac{11}{13}).
These are the basic problems on probability with playingcards.You might like these
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