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.

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:

(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

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

Therefore, probability of getting ‘2’ ofspade

Number of favorable outcomesP(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 outcomesP(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 outcomesP(C) = Total number of possible outcome = 2/52 = 1/26

(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 outcomesP(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 outcomesP(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 outcomesP(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 outcomesP(G) = Total number of possible outcome = 6/52 = 3/26

(viii) a black card:

Cards of spades and clubs are black cards.

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 outcomesP(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 outcomesP(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 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 outcomesP(J) = Total number of possible outcome = 20/52 = 5/13

(xi) neither a spade nor a jack

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

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

Therefore, probability of getting ‘neithera spade nor a jack’

Number of favorable outcomesP(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

Therefore, probability of getting ‘neithera heart nor a red king’

Number of favorable outcomesP(L) = Total number of possible outcome = 38/52 = 19/26

2. A card is drawn at 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 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, .

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= 44

Therefore, by definition, P(F) = $$\frac{44}{52}$$

= $$\frac{11}{13}$$.