# Playing Cards Probability

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 ♣.

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 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) = Number of Favourable Outcomes for the Event ETotal Number of Possible Outcomes

Number of Favourable Outcomes for the Event ETotal Number of Possible Outcomes

= 620

620

= 310

310.

(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) = Number of Favourable Outcomes for the Event FTotal Number of Possible Outcomes

Number of Favourable Outcomes for the Event FTotal Number of Possible Outcomes

= 620

620

= 310

310.

**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) = 452

452

= 113

113.

(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) = 4452

4452

= 1113

1113.

These are the basic problems on *probability* with **playing
cards.**