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Card-Based Probability

Introduction

कई probability questions में 52 playing cards की standard deck का उपयोग किया जाता है। Cards की structure (suits, colors और values) को समझने से आप किसी specific card या combination को निकालने की probability आसानी से find कर सकते हैं।

यह pattern इसलिए important है क्योंकि card-based problems exams में बहुत आते हैं और इनमें हमेशा fixed total 52 cards में से favourable outcomes पहचानने की जरूरत होती है।

Pattern: Card-Based Probability

Pattern

Main idea यह है कि favourable cards को total cards (52) से divide किया जाए।

उपयोग किया जाने वाला formula:
P(E) = (Number of favourable cards) / 52

Step-by-Step Example

Question

52 cards की well-shuffled deck से एक card निकाला जाता है। Red card आने की probability क्या है?

Solution

  1. Step 1: Total outcomes पहचानें

    Deck में total cards = 52.

  2. Step 2: Favourable outcomes पहचानें

    दो red suits होते हैं - Hearts और Diamonds। हर suit में 13 cards होते हैं, इसलिए total red cards = 13 + 13 = 26.

  3. Step 3: Formula लागू करें

    P(red card) = 26 / 52 = 1/2.

  4. Final Answer:

    1/2.

  5. Quick Check:

    26 red + 26 black = 52 cards → probabilities का sum = 1 ✅

Quick Variations

1. Face card (Jack, Queen, King) निकालने की probability।

2. किसी specific suit का card निकालने की probability।

3. Numbered card (2-10) निकालने की probability।

4. किसी खास card (जैसे Ace of Spades) की probability।

Trick to Always Use

  • Step 1: याद रखें total cards = 52 (13 प्रति suit × 4 suits)।
  • Step 2: Structure पहचानें: 26 red (Hearts & Diamonds), 26 black (Clubs & Spades)।
  • Step 3: पहले favourable cards count करें, फिर 52 से divide करें।

Summary

Summary

Card-Based Probability pattern में:

  • Total cards = 52.
  • हर suit (Hearts, Diamonds, Clubs, Spades) में 13 cards होते हैं।
  • Red suits → Hearts & Diamonds; Black suits → Clubs & Spades।
  • Probability = Favourable / 52.
  • Divide करने से पहले favourable outcomes हमेशा सही count करें।

Practice

(1/5)
1. One card is drawn at random from a standard 52-card deck. What is the probability that the card is an Ace?
easy
A. 1/13
B. 1/52
C. 1/4
D. 1/26

Solution

  1. Step 1: Identify total outcomes

    Total cards in deck = 52.

  2. Step 2: Count favourable outcomes

    There are 4 Aces (one per suit) → favourable = 4.

  3. Step 3: Apply formula

    P(Ace) = 4 / 52 = 1/13.

  4. Final Answer:

    1/13 → Option A.

  5. Quick Check:

    Divide numerator and denominator by 4: 4/52 = 1/13 ✅
Hint: Remember there are 4 of any specific rank in a 52-card deck.
Common Mistakes: Confusing a single suit Ace (1/52) with all Aces (4/52).
2. A single card is drawn from a well-shuffled deck. What is the probability that it is a Club?
easy
A. 1/2
B. 1/4
C. 1/13
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Count favourable outcomes

    Each suit (Clubs) has 13 cards → favourable = 13.

  3. Step 3: Apply formula

    P(Club) = 13 / 52 = 1/4.

  4. Final Answer:

    1/4 → Option B.

  5. Quick Check:

    Four suits → probability for any one suit = 1/4 ✅
Hint: There are 4 suits; probability of any specific suit = 1/4.
Common Mistakes: Using 26 (half the deck) by mistake when asked for one suit.
3. One card is drawn from a standard deck. What is the probability that the card is a numbered card (2 through 10)?
easy
A. 9/13
B. 3/13
C. 7/13
D. 1/2

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Count favourable outcomes

    Ranks 2-10 = 9 ranks per suit × 4 suits = 9 × 4 = 36 favourable cards.

  3. Step 3: Apply formula

    P(numbered card) = 36 / 52 = 9/13.

  4. Final Answer:

    9/13 → Option A.

  5. Quick Check:

    36/52 reduces by dividing by 4 → 9/13 ✅
Hint: Multiply number of ranks by 4 to get total of that rank-type in deck.
Common Mistakes: Counting Ace as a numbered card or forgetting that 10 is included.
4. A single card is drawn. What is the probability that it is a red face card (Jack, Queen, or King of Hearts or Diamonds)?
medium
A. 3/13
B. 1/13
C. 3/26
D. 1/4

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Count favourable outcomes

    Face cards per suit = 3 (J, Q, K). Two red suits (Hearts, Diamonds) → favourable = 3 × 2 = 6.

  3. Step 3: Apply formula

    P(red face card) = 6 / 52 = 3/26.

  4. Final Answer:

    3/26 → Option C.

  5. Quick Check:

    6/52 simplifies by 2 → 3/26 ✅
Hint: Count face cards per suit then multiply by number of suits needed.
Common Mistakes: Counting all face cards (12) instead of only the red ones (6).
5. One card is drawn at random. What is the probability that the card is either an Ace or a Heart?
medium
A. 1/3
B. 3/13
C. 5/13
D. 4/13

Solution

  1. Step 1: Identify total outcomes

    Total cards = 52.

  2. Step 2: Count favourable outcomes (Ace or Heart)

    Number of Aces = 4. Number of Hearts = 13. Intersection (Ace of Hearts) counted twice, so subtract 1. Total favourable = 4 + 13 - 1 = 16.

  3. Step 3: Apply formula

    P(Ace or Heart) = 16 / 52 = 4/13.

  4. Final Answer:

    4/13 → Option D.

  5. Quick Check:

    Alternative: 4/52 + 13/52 - 1/52 = 16/52 = 4/13 ✅
Hint: Use inclusion-exclusion: P(A or B) = P(A) + P(B) - P(A and B).
Common Mistakes: Adding counts without subtracting the overlap (Ace of Hearts).

Mock Test

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