Probability Questions Cards, Dice, Coins & Bags

Probability Questions Cards, Dice, Coins & Bags

Probability in Deck of Cards, Dice, and Other Games – Notes for Competitive Exams

Introduction

Probability is a key topic in Quantitative Aptitude and Reasoning sections of competitive exams like SSC, Banking, Railway, and Defence.

This topic helps calculate the likelihood of events when using:

  • Dice
  • Coins
  • Decks of cards
  • Balls in bags

These are frequently asked in exams because they are easy to understand with proper practice.

Key Formulas

  1. Probability of an Event (E):

P(E)=Number of favorable outcomesTotal number of outcomesP(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}P(E)=Total number of outcomesNumber of favorable outcomes​

  1. Complementary Probability:

P(Not E)=1−P(E)P(\text{Not E}) = 1 – P(E)P(Not E)=1−P(E)

  1. Addition Rule (Mutually Exclusive Events):

P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)P(A or B)=P(A)+P(B)

  1. Multiplication Rule (Independent Events):

P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)P(A and B)=P(A)×P(B)

Probability in Dice

  • A standard die has 6 faces (1–6).
  • Probability of rolling a number n: 1/61/61/6
  • Probability of rolling an even number: 3/6=1/23/6 = 1/23/6=1/2
  • Probability of rolling a number greater than 4: 2/6=1/32/6 = 1/32/6=1/3

Example:
Probability of rolling a 3 or 5: 1/6+1/6=1/31/6 + 1/6 = 1/31/6+1/6=1/3

Probability in Coins

  • A coin has 2 outcomes: Head or Tail.
  • Probability of Head or Tail: 1/21/21/2
  • Probability of 2 heads in 2 tosses: 1/41/41/4
  • Probability of at least one head in 2 tosses: 3/43/43/4

Probability in Deck of Cards

  • A standard deck has 52 cards: 4 suits × 13 cards
  • Probability of drawing a:
    • Ace: 4/52 = 1/13
    • King or Queen: 8/52 = 2/13
    • Heart: 13/52 = 1/4
    • Spade or red card: (13+26)/52 = 39/52 = 3/4

Probability in Balls/Objects in a Bag

  • A bag has different colored balls (e.g., 5 red, 3 blue).
  • Probability of drawing a red ball: 5/8
  • Probability of not drawing blue: 5/8

Example:
Bag has 4 white, 3 black balls. Probability of drawing black or white = (3+4)/7 = 1

Tips & Tricks

  • Count total outcomes carefully
  • Identify favorable outcomes
  • Use addition and multiplication rules wisely
  • Use complementary probability for “not happening” problems
  • Practice dice, coin, card, and bag-based questions regularly

Examples

  1. Dice: Probability of getting a prime number = 3/6 = 1/2 (numbers 2, 3, 5)
  2. Cards: Probability of drawing a red King = 2/52 = 1/26
  3. Coins: Probability of getting exactly 1 tail in 2 tosses = 1/2

Probability Questions – Cards, Dice, Coins, and Bags

Q1. A single die is rolled. What is the probability of getting a 4?

Q2. A coin is tossed. Probability of getting a head?

Q3. Two dice are rolled. Probability of getting a double 6?

Q4. A card is drawn from a standard deck. Probability of getting a King?

Q5. A bag contains 5 red and 7 blue balls. Probability of picking a red ball?

Q6. Probability of getting an even number when a die is rolled?

Q7. A coin is tossed twice. Probability of getting two heads?

Q8. Probability of getting at least one head in two coin tosses?

Q9. A deck of 52 cards: Probability of drawing a heart?

Q10. Probability of drawing an Ace or a King from a deck?

Q11. Probability of rolling a number greater than 4 on a die?

Q12. A bag contains 4 white and 6 black balls. Two balls are drawn without replacement. Probability both are black?

Q13. Probability of drawing a red card or a black King from a deck?

Q14. Probability of rolling an odd number on a die?

Q15. Probability of getting exactly one tail when two coins are tossed?

Q16. Probability of getting a number divisible by 3 when a die is rolled?

Q17. Probability of drawing a spade from a standard deck?

Q18. A bag has 3 green, 2 yellow, and 5 red balls. Probability of drawing a green ball?

Q19. Probability of getting at least one six when two dice are rolled?

Q20. Probability of drawing a face card from a standard deck of 52 cards?

Q21. Probability of getting no heads when three coins are tossed?

Q22. Probability of getting a sum of 7 when two dice are rolled?

Q23. Probability of drawing a number card (2–10) from a deck?

Q24. Probability of drawing a black card or a Queen from a deck?

Q25. Probability of getting a prime number when a single die is rolled?

Answer

Probability – Answers (Cards, Dice, Coins, Bags)

Q1. 1/6
Q2. 1/2
Q3. 1/36
Q4. 4/52 = 1/13
Q5. 5/12
Q6. 3/6 = 1/2
Q7. 1/4
Q8. 3/4
Q9. 13/52 = 1/4
Q10. (4+4)/52 = 8/52 = 2/13
Q11. 2/6 = 1/3
Q12. (6/10) × (5/9) = 1/3
Q13. (26+4)/52 = 30/52 = 15/26
Q14. 3/6 = 1/2
Q15. 2/4 = 1/2
Q16. 2/6 = 1/3
Q17. 13/52 = 1/4
Q18. 3/10
Q19. 11/36
Q20. 12/52 = 3/13
Q21. 1/8
Q22. 6/36 = 1/6
Q23. 36/52 = 9/13
Q24. (26 + 4 – 1)/52 = 29/52 (subtract overlap if needed)
Q25. 3/6 = 1/2