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Card-Based Probability

Introduction

பல probability கேள்விகளில், 52 playing cards கொண்ட standard deck பயன்படுத்தப்படுகிறது. card structure-ஐ (suits, colors, values) புரிந்துகொண்டால், குறிப்பிட்ட cards அல்லது combinations-ஐ draw செய்வதற்கான probability-ஐ எளிதாக கணக்கிடலாம்.

exams-ல் card-based problems அடிக்கடி வருவதால் இது முக்கியம்; 52 cards என்ற நிலையான total-இல் இருந்து favourable outcomes-ஐ தெளிவாக அடையாளம் காண வேண்டும்.

Pattern: Card-Based Probability

Pattern

முக்கிய கருத்து: deck-இல் உள்ள favourable cards எண்ணிக்கையும் total cards (52) எண்ணிக்கையும் உள்ள விகிதத்தை கண்டறிதல்.

பயன்படுத்தப்படும் Formula:
P(E) = (Number of favourable cards) / 52

Step-by-Step Example

Question

நன்றாக shuffle செய்யப்பட்ட 52 cards கொண்ட deck-இலிருந்து ஒரு card draw செய்யப்படுகிறது. red card வருவதற்கான probability என்ன?

Solution

  1. Step 1: total outcomes-ஐ கண்டறியவும்

    deck-இல் உள்ள total cards = 52.

  2. Step 2: favourable outcomes-ஐ கண்டறியவும்

    இரண்டு red suits உள்ளன - Hearts மற்றும் Diamonds. ஒவ்வொரு suit-இலும் 13 cards இருப்பதால், total red cards = 13 + 13 = 26.

  3. Step 3: formula-ஐ பயன்படுத்தவும்

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

  4. Final Answer:

    1/2.

  5. Quick Check:

    26 red + 26 black = 52 cards → probabilities கூட்டுத்தொகை 1 ஆகிறது ✅

Quick Variations

1. face card (Jack, Queen, King) draw செய்வதற்கான probability.

2. குறிப்பிட்ட suit-இன் card draw செய்வதற்கான probability.

3. numbered card (2-10) draw செய்வதற்கான probability.

4. குறிப்பிட்ட card (எ.கா., Ace of Spades) draw செய்வதற்கான probability.

Trick to Always Use

  • Step 1: total cards = 52 (ஒவ்வொரு suit-க்கும் 13 × 4 suits) என்பதை நினைவில் வைத்துக்கொள்ளவும்.
  • Step 2: structure-ஐ தெரிந்து கொள்ளவும்: 26 red (Hearts & Diamonds), 26 black (Clubs & Spades).
  • Step 3: முதலில் favourable cards-ஐ எண்ணி, பின்னர் 52-ஆல் divide செய்யவும்.

Summary

Summary

Card-Based Probability pattern-இல்:

  • deck-இல் total cards = 52.
  • ஒவ்வொரு suit (Hearts, Diamonds, Clubs, Spades)-இலும் 13 cards உள்ளன.
  • Red suits → Hearts & Diamonds; Black suits → Clubs & Spades.
  • Probability = Favourable / 52.
  • divide செய்வதற்கு முன் favourable outcomes-ன் எண்ணிக்கையை எப்போதும் சரிபார்க்கவும்.

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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