Playing Cards Probability and Face Card Combinations for CBSE Class 10

Among all the standard probability models used in mathematics examinations, none is more popular with board paper setters than the classic pack of $52$ playing cards. A deck of cards is a masterclass in combinatorial symmetry: perfectly divided into two colours, four suits, and thirteen sequential ranks.

However, for students who do not regularly play card games, terms like "spades", "face cards", "honour cards", or "knaves/jacks" can feel like a foreign language. In CBSE Class 10 Mathematics, Chapter 14 (Probability), mastering the complete taxonomy of a 52-card deck guarantees an effortless 3 to 4 marks in your board exam.

What You Will Learn

• Complete architectural taxonomy of a standard deck of $52$ playing cards
• The $2$ colours: Red ($26$) vs. Black ($26$)
• The $4$ suits: Spades ($\spadesuit$), Clubs ($\clubsuit$), Hearts ($\heartsuit$), and Diamonds ($\diamondsuit$)
• What are Face Cards? (Kings, Queens, and Jacks)
• The status of the Ace: Why an Ace is NOT a face card!
• Solving standard board exam questions (red face cards, kings, spades)
• Card removal problems: How the sample space contracts when cards are discarded
• Presentation formats and common traps

1. The Anatomy of a Standard 52-Card Deck

A standard deck contains $52$ cards (excluding jokers), arranged in a symmetrical hierarchy:

                                  A Standard Deck of 52 Cards
                                               |
       +---------------------------------------+---------------------------------------+
       |                                                                               |
26 RED CARDS                                                                    26 BLACK CARDS
       |                                                                               |
   +---+---+                                                                       +---+---+
   |       |                                                                       |       |
13 Hearts  13 Diamonds                                                          13 Spades  13 Clubs
 ( ♥ )      ( ♦ )                                                                ( ♠ )      ( ♣ )

The 13 Denominations in Every Suit:

Every one of the four suits contains the exact same sequence of $13$ cards:

1. Ace ($A$): 1 per suit $\implies$ $4$ Aces in total (2 Red, 2 Black).
2. Numbered Cards (Pips): $2, 3, 4, 5, 6, 7, 8, 9, 10$ ($9$ cards per suit $\implies$ $36$ Numbered cards in total).
3. Face Cards (Court Cards): Jack ($J$), Queen ($Q$), and King ($K$) ($3$ cards per suit $\implies$ $12$ Face cards in total).

2. What is a Face Card? (CBSE Core Focus)

A face card (or court card) is a card that displays a physical illustration of a human face:

                            The 12 Face Cards
                                    |
       +----------------------------+----------------------------+
       |                                                         |
6 RED FACE CARDS                                          6 BLACK FACE CARDS
- 2 Kings (Heart K, Diamond K)                            - 2 Kings (Spade K, Club K)
- 2 Queens (Heart Q, Diamond Q)                           - 2 Queens (Spade Q, Club Q)
- 2 Jacks (Heart J, Diamond J)                            - 2 Jacks (Spade J, Club J)

The Single Most Common Card Trap: <u>An ACE IS NOT A FACE CARD! An Ace has no human face illustrated on it; it simply carries the letter 'A' and a single central suit symbol. The only face cards are KINGS, QUEENS, and JACKS ($4 imes 3 = 12 ext{ cards}$)!</u>

3. Card Taxonomy Summary Table

4. Solved CBSE Board Examination Problems

Solved Example 1: Standard Card Probability (NCERT Classic)

Problem: 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

Solution: Total number of cards in deck: $n(S) = 52$.

1. (i) A king of red colour: There are two red suits (Hearts and Diamonds), each containing 1 king: Favourable outcomes $= 2$ (King of Hearts, King of Diamonds). $P(\text{Red King}) = \frac{2}{52} = \mathbf{\frac{1}{26}}$

2. (ii) A face card: There are 4 Kings, 4 Queens, and 4 Jacks: Favourable outcomes $= 4 + 4 + 4 = 12$. $P(\text{Face Card}) = \frac{12}{52} = \mathbf{\frac{3}{13}}$

3. (iii) A red face card: There are 3 face cards in Hearts and 3 in Diamonds: Favourable outcomes $= 3 + 3 = 6$. $P(\text{Red Face Card}) = \frac{6}{52} = \mathbf{\frac{3}{26}}$

4. (iv) The jack of hearts: There is only ONE jack of hearts in the entire deck: $P(\text{Jack of Hearts}) = \mathbf{\frac{1}{52}}$

5. (v) A spade: There are 13 spades in the deck: $P(\text{Spade}) = \frac{13}{52} = \mathbf{\frac{1}{4}}$

6. (vi) The queen of diamonds: There is only ONE queen of diamonds in the deck: $P(\text{Queen of Diamonds}) = \mathbf{\frac{1}{52}}$

Solved Example 2: Card Removal Problem (NCERT Classic)

Problem: Five cards—the ten, jack, queen, king, and ace of diamonds—are well-shuffled 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?

Solution:

1. (i) Initial Draw:

   • Total cards available: $n(S) = \mathbf{5}$ (10, J, Q, K, A of diamonds).
   • There is 1 queen among the 5 cards: $P(\text{Queen}) = \mathbf{\frac{1}{5}}$

2. (ii) After Queen is Put Aside:

   • The queen has been removed and NOT replaced!
   • Remaining cards: $\{10, J, K, A\} \implies$ New total sample space is $n(S') = 4$!
   • (a) An ace: There is 1 ace among the remaining 4 cards: $P(\text{Ace}) = \mathbf{\frac{1}{4}}$
   • (b) A queen: There are ZERO queens remaining among the 4 cards (Impossible event!): $P(\text{Queen}) = \frac{0}{4} = \mathbf{0}$

5. Summary and Examination Tips

Exam Tip: In card removal questions (like Solved Example 2), PAY CAREFUL ATTENTION to the denominator! When cards are put aside, the total sample space shrinks ($52 \to 51$ or $5 \to 4$). Forgetting to reduce the denominator loses the entire mark!

Common Mistake: Counting Aces as face cards. Remember: There are 12 face cards, NOT 16!