**Playing Cards**

A pack of playing cards has 52 cards. There are 4 suits (spade, heart, diamond and club) each having 13 cards. There are two colours, red (heart and diamond) and black (spade and club) each having 26 cards.

In 13 cards of each suit, there are 3 face cards namely king, queen and jack so there are in all 12 face cards. Also, there are 16 honour cards, 4 of each suit namely ace, king, queen and jack.

Now, let us take an example related to playing cards. Have you seen a deck of playing cards? It consists of 52 cards which are divided into 4 suits of 13 cards each spades (♠), hearts (♥), diamonds (♦) and clubs (♣). Clubs and spades are of black colour, while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2. Kings, queens and jacks are called face cards

Ex1. A card is drawn at random from a deck of cards. Find the probability of getting the 3 of diamond.

solution:

The sample space S of this experiment in Ex1 is shown below

×

{A,2,3,4,5,6,7,8,9,10,J,Q,K}

Let E be the event “getting the 3 of diamond”. An examination of the sample space shows that there is one “3 of diamond” so that

*n*(E)=1 and

*n*(S)=52. Hence the probability of event E occuring is given by

*P*(E)=1/52.

Ex2. A card is drawn at random from a deck of cards. Find the probability of getting a queen.

solution:

The sample space S of the experiment in question 7 is shwon above (see Ex1)

Let E be the event “getting a Queen”. An examination of the sample space shows that there are 4 “Queens” so that *n*(E)=4 and *n*(S)=52. Hence the probability of event E occuring is given by *P*(E)=4/52=1/13.

Ex3. A card is drawn at random from a deck of cards. Find the probability of getting the King of hearts.

Solution:

A deck contains 52 cards.

Ex4. One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will

(i) be an ace,

(ii) not be an ace.

solutions:

Well-shuffling ensures

*equally likely*outcomes.

(i) There are 4 aces in a deck. Let E be the event ‘the card is an ace’.

The number of outcomes favorable to E is equal to 4.

The number of possible outcomes =52 (Why?)

Therefore,

*P*(E)=4/52=1/13

(ii) Let F be the event ‘card drawn is not an ace’.

The number of outcomes favorable to the event F is 52-4=48 (Why?)

The number of possible outcomes =52

Therefore, *P*(F)=48/52=12/13

Remark: Note that F is nothing but Ē. Therefore, we can also calculate *P*(F) as follows:

Ex5. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red colour

(ii) a face card

(iii) a red face card

(iv) the jack of hearts

(v) a spade

(vi) the queen of diamonds

solutions:

Total number of cards in a well -shuffled deck =52

(i) Total number of kings of red colour =2

Number of favorable outcomes

*P*(getting a king of red colour)=2/52=1/26

(ii) Total number of face cards =12

*P*(getting a face card)=12/52=3/13

(iii) Total number of red face cards =6

*P*(getting a red face card)=6/52=3/26

(iv) Total number of Jack of hearts =1

*P*(getting a Jack of hearts)=1/52

(v) Total number of spade cards =13

*P*(getting a spade card)=13/52=¼

(vi) Total number of queen of diamonds =1

*P*(getting a queen of diamond)=1/52

Ex6. Five cards — the ten, jack, queen, king and ace of diamonds, are wellshuffled with their face downwards. One card is then picked up at random.

(i) What is the probability that the card is the queen?

(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?

solutions:

(i) Total number of cards =5; total number of queens =1

*P*(getting a queen)=⅕

(ii) When the queen is drawn and put aside, the total number of remaining cards will be 4.

(a) Total number of aces =1

*P*(getting an ace)=¼

(b) As queen is already drawn, therefore, the number of queens will be 0.

*P*(getting a queen)=0/4=0

Ex7. From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn will:

(i) be a black card.

(ii) not be a red card.

(iii) be a red card.

(iv) be a face card.

(v) be a face card of red colour.

A deck contains 52 cards which are divided into 4 suits of 13 cards each spades (♠), hearts (♥), diamonds (♦) and clubs (♣). Clubs and spades are of black colour, while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2. Kings, queens and jacks are called face cards.

solutions:

Total number of cards =52

Total number of outcomes =*n*(S)=52

There are 13 cards of each type. The cards of heart and diamond are red in colour.

Spade and diamond are black. So, there are 26 red cards and 26 black cards.

(i) Number of black cards in a deck =26

*n*(E)= favourable outcomes for the event of drawing a black card =26

Probability of drawing a black card

(ii) Number of red cards in a deck =26

Therefore, number of non-red cards =52-26=26

*n*(E)= favourable outcomes for the event of not drawing a red card =26

Probability of not drawing a red card

(iii) Number of red cards in a deck =26

*n*(E)= favourable outcomes for the event of drawing a red card =26

Probability of drawing a red card

(iv) There are 52 cards in a deck of cards, and 12 of these cards are face cards (4 kings, 4 queens, and 4 jacks).

*n*(E)=12

Probability of drawing a face card

(v) There are 26 red cards in a deck. and 6 of these cards are face cards (2 kings. 2 queens. and 2 jacks).

*n*(E)=6

Probability of drawing a red face card

Ex8. A card is drawn from a well shuffled pack of 52 cards. Find the probability that the card drawn is:

(i) a spade (v) Jack or queen

(ii) a red card (vi) ace and king

(iii) a face card (vii) a red and a king

(iv) 5 of heart or diamond (viii) a red or a king

solutions:

Number of possible outcomes when card is drawn from pack of 52 cards, *n*(S)=52.

(i) Number of spade cards =¼*n*(S), E= event of drawing a spade, *n*(E)=¼*n*(S)

Probability of drawing a spade

(ii) Number of red cards (hearts+diamonds) =½

*n*(S), E= event of drawing a red card,

*n*(E)=½

*n*(S)

Probability of drawing a red card –

(iii) Number of face cards (4 kings+4 queens+4 Jacks),

*n*(E)=12

Probability of drawing a face card

(iv) E= event of drawing a 5 of heart, or of diamond ={5H,5D},

*n*(E)=2

Probability of drawing a 5 of heart or of diamond

(v) E= event of drawing a jack or a queen ={JH,JS,JD,JC,QH,QS,QD,QC},

*n*(E)=8

Probability of drawing a jack or a queen

(vi) A card cannot be both an ace as well as a king.

E= event of drawing an ace and a king,

*n*(E)=0.

Probability of drawing an ace and a king =

*n*(E)/

*n*(S) =0/52=0 (vii) E= event of drawing a red and a king ={KR,KD}.

Probability of drawing a red and a king =

*n*(E)/

*n*(S) =2/52=1/26

(viii) E= event of drawing a red card or a red king =26 red cards and 2 black kings.

*n*(E)=26+2=28.

Probability of drawing a red or a king

*n*(E)/

*n*(S) =28/52=7/13

Ex9. A card is drawn from a pack of 52 cards. Find the probability that the card drawn is:

(i) a red card (ii) a black card (iii) a spade (iv) an ace

(v) a black ace (vi) ace of diamonds (vii) not a club

(viii) a queen or a jack

solutions:

Number of possible outcomes when card is drawn from pack of 52 cards, *n*(S)=52

(i) Number of red cards (hearts+diamonds) *n*(E)=26

Probability of drawing a red card =*n*(E)/*n*(S) =26/52=½

(ii) Number of black cards (spade+dubs) , *n*(E)=26

Probability of drawing a black card =*n*(E)/*n*(S) =26/52=½

(iii) Number of spade cards *n*(E)=13

Probability of drawing a spade =*n*(E)/*n*(S) =13/52=¼

(iv) Number of ace cards *n*(E)=4

Probability of drawing an ace =*n*(E)/*n*(S) =4/52=1/13

(v) Number of black ace cards *n*(E)=2

Probability of drawing a black ace =*n*(E)/*n*(S) =2/52=1/26

(vi) There is only one ace of diamonds. E= event of drawing an ace of diamonds, *n*(E)=1.

Probability of drawing an ace of diamonds =*n*(E)/*n*(S) =1/52

(vii) Number of club cards *n*(E)=¼×52

Probability of drawing a club card

Probability of not drawing a club card 1-¼=¾

(viii) E= event of drawing a jack or a queen,

*n*(E)=4×2=8

Probability of drawing a jack or a queen =

*n*(E)/

*n*(S) =8/52=2/13

Ex10. From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn is:

(i) a face card

(ii) not a face card

(iii) a queen of black card

(iv) a card with number 5 or 6

(v) a card with number less than 8

(vi) a card with number between 2 and 9

solutions:

A deck contains 52 cards which are divided into 4 suits of 13 cards each spades (♠)(S), hearts (♥)(H), diamonds (♦)(D) and clubs (♣)(C).

Total number of possible outcomes =*n*(S)=52

(i) No. of face cards in a deck of 52 cards =12 (4 kings, 4 queens and 4 jacks)

Event of drawing a face cards =E= (4 kings. 4 queens and 4 jacks) =*n*(E)=12

Probability of drawing a face card

(ii) Probability of not drawing a face card =1- probability of drawing a face card

Probability of not drawing a face card

(iii) Event of drawing a queen of black color E={Q(spade), Q(club)},

*n*(E)=2

Probability of drawing a queen of black color

(iv) Event of drawing a card with number 5 or 6 =E={5H, 5D, 55, 5C, 6H, 6D, 65, 6C},

*n*(E)=8

Probability of drawing a card with number 5 or 6

(v) Numbers less than 8 ={2,3,4,5,6,7}

Event of drawing a card with number less than 8 =E={6H cards, 6D cards. 6S cards. 6C cards},

*n*(E)=24

Probability of drawing a card with number less than 8

(vi) Number between 2 and 9 ={3,4,5,6,7,8}

Event of drawing a card with number between 2 and 9 =E={6H cards, 6D cards. 6S cards, 6C cards}

Probability of drawing a card with number between 2 and 9

Ex11. All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting:

(i) a black face card

(ii) a queen

(iii) a black card

solutions:

Total number of cards =52

3 face cards of spades are removed

Remaining cards =52-3=49= number of possible outcomes *n*(S) =49

(i) Number of black face cards left =3 face cards of club

Event of drawing a black face card *n*(E)=3

Probability of drawing a black face card =*n*(E)/*n*(S) =3/49

(ii) Number of queen cards left =3

Event of drawing a black face card *n*(E)=3

Probability of drawing a queen card =*n*(E)/*n*(S) =3/49

(iii) Number of black cards left =23 cards (13 club 10 spade)

Event of drawing a black card *n*(E)=23

Probability of drawing a black card =*n*(E)/*n*(S) =23/49

Ex12. One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting:

(i) a queen of red color

(ii) a black face card

(iii) the jack or the queen of the hearts

(iv) a diamond

(v) a diamond or a spade

solutions:

Total possible outcomes =52

(i) Number queens of red color =2

Number of favorable outcomes =2

*P*(queen of red color) =2/52=1/26

(ii) Number of black cards =26

Number of black face cards =6

Number of favorable outcomes =6

*P*(black face card) =6/52=3/26

(iii) Favorable outcomes for jack or queen of hearts =1 jack+1 queen

Number of favorable outcomes =2

*P*(jack or queen of hearts) =2/52=1/26

(iv) Number of favorable outcomes for a diamond =13

Number of favorable outcomes =13

*P*(getting a diamond) =13/52=¼

(v) Number of favorable outcomes for a diamond or a spade =13+13=26

Number of favorable outcomes =26

*P*(getting a diamond or a spade) =26/52=½

Ex13. From a deck of 52 cards, all the face cards are removed and then the remaining cards are shuffled. Now one card is drawn from the remaining deck. Find the probability that the card drawn is:

(i) a black card

(ii) 8 of red color

(iii) a king of black color

solutions:

There are 12 face cards in a deck.

Therefore, possible number of outcomes =52-12=40

(i) number of favorable outcomes for black cards =26 cards -6 face cards =20

*P*(a black card) =20/40=½

(ii) number of favorable outcomes for 8 of red color =2

*P*(getting a card with 8 of red color) =2/40=1/20

(iii) Since all face cards are removed

Number of favorable outcomes for a king of black color =0

*P*(getting a king of black color) =0/40=0

Ex14. Seven cards:- the eight, the nine, the ten, jack, queen, king and ace of diamonds are well shuffled. One card is then picked up at random.

(i) What is the probability that the card drawn is the eight or the king?

(ii) If the king is drawn and put aside, what is the probability that the second card picked up is:

a) an ace? b) a king?

solutions:

Total number of possible outcomes =7

(i) Number of favorable outcomes for the card is 8 or the king =2

*P*(card is 8 or the king) =2/7

(ii) a) If a king is drawn and put aside. then total possible outcomes =6

Number of favorable outcomes for an ace =1

*P*(card is an ace) =⅙

b) Now, for second pick number of king =0

Number of favorable outcomes for a king =0

*P*(card is a king) =0/6=6

Ex15. From a pack of 52 playing cards, all cards whose numbers are multiples of 3 are removed. A card is now drawn at random. What is the probability that the card drawn is

(i) A face card (King, Jack or Queen)

(ii) An even numbered red card?

solutions:

Number of total cards =52

Cards removed of 4 colours of multiples of 3 are {3,6,9}

4×3=12

Remaining cards =52-12=40.

(i) Number of face cards {J,Q,K} =12 cards

The probability *P*=12/40=3/10=0.3

(ii) An even number red cards {2,4,6,8}, 4 cards, the number of that cards =2×4=8

The probability *P*=8/40=2/10=0.2

Ex16. One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting:

(i) a king of red suit (ii) a face card (iii) a red face card

(iv) a queen of black suit (v) a jack of hearts (vi) a spade

solutions:

Total number of possible outcomes *n*(S)=52 (52 cards)

(i) E→ event of getting a king of red suit

Number of favorable outcomes *n*(E)=2 {king heart & king of diamond}

(ii) E→ event of getting face card

Number of favorable outcomes

*n*(E)=12 (4 kings, 4 queens & 4 jacks)

(iii) E→ event of getting red face card

Number of favorable outcomes

*n*(E)=6 (kings, queens, jacks of hearts & diamonds)

(iv) E→ event of getting a queen of black suit

Number of favorable outcomes

*n*(E)=2 (spade queen and club one)

(v) E→ event of getting a jack of hearts

Number of favorable outcomes

*n*(E)=1

It is very difficult to get each specific card.

(vi) E→ event of getting a spade

Number of favorable outcomes

*n*(E)=¼×

*n*(S)

Ex17. Five cards: ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random.

(i) What is the probability that the card is a queen?

(ii) If a king is drawn first and put aside, what is the probability that the second card picked up is the ace?

solutions:

Total number of possible outcomes =*n*(E)=5 {5 cards}

(i) E→ event of drawing queen

(ii) When king is drawn and put aside, total number of remaining cards 4. Total number of possible outcomes

*n*(S)=4.

E→ event of drawing ace card

Number of favorable outcomes

*n*(S)=1 (1 ace card)

Ex18. From a pack of 52 playing cards Jacks, queens, kings and aces of red colour are removed. From the remaining, a card is drawn at random. Find the probability that the card drawn is

(i) A black queen (ii) A red card (iii) A black jack

(iv) a picture card (Jacks, queens and kings are picture cards)

solutions:

Total number of cards =52

All jacks, queens & kings, aces of red colour are removed.

Total number of possible outcomes

*n*(S)=52-2-2-2-2=44 (remaining cards)

(i) E→ event of getting a black queen

Number of favorable outcomes

*n*(E)=2, (queen of spade & club).

(ii) E→ event of getting a red card

Number of favorable outcomes

*n*(E)=26-8=18 (after removing jacks, queens, kings, aces from total red cards)

(iii) E→ event of getting a black jack

Number of favorable outcomes

*n*(E)=2 (jack of club & spade)

(iv) E→ event of getting a picture card

Number of favorable outcomes

*n*(E)=6 (2 jacks, 2 kings & 2 queens of black colour)

Ex19. The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of

(i) heart (ii) queen (iii) clubs.

solutions:

Total number of remaining cards *n*(S)=52-3=49

(i) E→ event of getting hearts. Heart is a suit of 4 ones of a deck, *n*(E)=¼×52=13

(ii) E→ event of getting queen

Number of favorable outcomes

*n*(E)=4-1=3 (Since queen of clubs is removed)

*P*(E)=3/49

(iii) E→ event of getting clubs

Number of favorable outcomes

*n*(E)=13-3=10 (Since 3 club cards are removed)

*P*(E)=10/49

(a) How many points are there in the sample space?

(b) Calculate the probability that the card is an ace of spades.

(c) Calculate the probability that the card is (i) an ace (ii) black card.

Solutions:

(a) When a card is selected from a pack of 52 cards, the number of possible outcomes is 52 i.e., the sample space contains 52 elements.

Therefore, there are 52 points in the sample space.

(b) Let A be the event in which the card drawn is an ace of spades. Accordingly,

*n*(A)=1

(c). (i). Let E be the event in which the card drawn is an ace.

Since there are 4 aces in a pack of 52 cards,

*n*(E)=4

(ii). Let F be the event in which the card drawn is black.

Since there are 26 black cards in a pack of 52 cards,

*n*(F)=26

Ex21. A card is drawn at random from a pack of 52 cards. Find the probability that 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

Sol:

Total number of outcomes *n*(S)=52 (52 cards)

(i) E→ event of getting a black king.

Number of favorable outcomes *n*(E)=2 (king of spades & king of clubs).

The probability is

(ii) E→ event of getting either a black card or a king.

Number of favorable outcomes

*n*(E)=26+2=28 (13 spades, 13 clubs, king of hearts & diamonds).

The probability is

(iii) E→ event of getting black & a king.

Number of favorable outcomes

*n*(E)=2 (king of spades & clubs).

The probability is

(iv) E→ event of getting a jack, queen or a king.

Number of favorable outcomes

*n*(E)=4+4+4=12 (4 jacks, 4 queens & 4 kings).

The probability is

(v) E→ event of getting neither a heart nor a king.

Number of favorable outcomes =52-13-3=36 (since we have 13 hearts, 3 kings each of spades, clubs & diamonds)

The probability is

(vi) E→ event of getting spade or an all.

Number of favorable outcomes

*n*(E)=13+3=16 (13 spades & 3 aces each of hearts, diamonds & clubs)

The probability is

(vii) E→ event of getting neither an ace nor a king.

Number of favorable outcomes

*n*(E)=52-4-4=44 (Since we have 4 aces & 4 kings).

The probability is

(viii) E→ event of getting neither a red card nor a queen.

Number of favorable outcomes

*n*(E)=52-26-2=24 (Since we have 26 red cards of hearts & diamonds & 2 queens each of heart & diamond).

The probability is

(ix) E→ event of getting card other than an ace.

Number of favorable outcomes

*n*(E)=52-4=48 (since we have 4 ace cards).

The probability is

(x) E→ event of getting a ten.

Number of favorable outcomes

*n*(E)=4 (each ten of 4 suits). The probability is

(xi) E→ event of getting a spade. Spade is a suit of four ones.

Number of favorable outcomes

*n*(E)=¼×

*n*(S). The probability is

(xii) E→ event of getting a black card. Black takes a half of the entire deck as many as red.

Number of favorable outcomes

*n*(E)=½×

*n*(S). The probability is

(xiii) E→ event of getting a club 7.

*n*(E)=1. The probability is

(xiv) E→ event of getting a jack (J).

Number of favorable outcomes

*n*(E)=4 {4 jack cards}. The probability is

(xv) E→ event of getting spade ace.

*n*(E)=1 The probability is

(xvi) E→ event of getting a queen (Q).

Number of favorable outcomes

*n*(E)=4 {4 queens}. The probability is

(xvii) E→ event of getting a heart. Heart is a suit of four ones.

*n*(E)=¼×

*n*(S). The probability is

(xviii) E→ event of getting a black card. Red takes a half of the entire deck as many as black.

Number of favorable outcomes

*n*(E)=½×

*n*(S). The probability is