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30 Probability Questions and Answers

30 solved Probability questions with step-by-step explanations. Includes formulas, shortcuts, and tips for placement tests and competitive exams.

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Advanced probability questions for experienced learners. These medium-to-hard questions test deeper understanding and are ideal for those who have already mastered the basics. Each question includes a detailed explanation.

Key Formulas

P(Event) = Favorable outcomes / Total possible outcomes For coins: Total outcomes = 2^n. For dice: Total outcomes = 6^n Independent events: P(A and B) = P(A) × P(B) Mutually exclusive: P(A or B) = P(A) + P(B) Not mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B) P(not A) = 1 - P(A)
Q1 Medium

From a 52-card deck, what is the probability of drawing a face card (J, Q, or K)?

  • A. 10/52
  • B. 12/52 ✓
  • C. 16/52
  • D. 13/52
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Each suit has J, Q, K so face cards = 3 x 4 = 12. Probability = 12/52 = 3/13.
Q2 Medium

A bag has 5 red and 5 blue balls. Probability of picking a red ball?

  • A. 10/5
  • B. 5/10 (1)
  • C. 5/10 ✓
  • D. 5/5
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 10. So probability = 5/10.
Q3 Medium

A fair die is rolled once. What is the probability of getting a prime number?

  • A. 2/6
  • B. 4/6
  • C. 3/6 ✓
  • D. 1/6
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Prime outcomes are 2, 3, 5 (3 outcomes) out of 6. Probability = 3/6 = 1/2.
Q4 Medium

A bag has 5 red and 3 blue balls. Probability of picking a red ball?

  • A. 5/8 ✓
  • B. 3/8
  • C. 8/5
  • D. 5/3
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 8. So probability = 5/8.
Q5 Medium

From a standard 52-card deck, what is the probability of drawing a Diamond?

  • A. 26/52
  • B. 1/4
  • C. 13/52 ✓
  • D. 1/2
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. There are 13 diamonds in 52 cards. Probability = 13/52 = 1/4.
Q6 Medium

A bag has 5 red and 2 blue balls. Probability of picking a red ball?

  • A. 5/7 ✓
  • B. 7/5
  • C. 2/7
  • D. 5/2
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 7. So probability = 5/7.
Q7 Medium

A bag has 6 red and 2 blue balls. Probability of picking a red ball?

  • A. 6/8 ✓
  • B. 8/6
  • C. 6/2
  • D. 2/8
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 6, total outcomes = 8. So probability = 6/8.
Q8 Medium

A bag has 3 red and 8 blue balls. Probability of picking a red ball?

  • A. 8/11
  • B. 3/8
  • C. 3/11 ✓
  • D. 11/3
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 3, total outcomes = 11. So probability = 3/11.
Q9 Medium

A bag has 5 red and 4 blue balls. Probability of picking a red ball?

  • A. 5/4
  • B. 9/5
  • C. 5/9 ✓
  • D. 4/9
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 9. So probability = 5/9.
Q10 Medium

A bag has 1 red and 2 blue balls. Probability of picking a red ball?

  • A. 1/3 ✓
  • B. 1/2
  • C. 3/1
  • D. 2/3
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 1, total outcomes = 3. So probability = 1/3.
Q11 Medium

A bag has 3 red and 3 blue balls. Probability of picking a red ball?

  • A. 6/3
  • B. 3/6 ✓
  • C. 3/6 (1)
  • D. 3/3
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 3, total outcomes = 6. So probability = 3/6.
Q12 Medium

A bag has 1 red and 7 blue balls. Probability of picking a red ball?

  • A. 8/1
  • B. 1/8 ✓
  • C. 7/8
  • D. 1/7
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 1, total outcomes = 8. So probability = 1/8.
Q13 Medium

A bag has 6 red and 7 blue balls. Probability of picking a red ball?

  • A. 13/6
  • B. 6/13 ✓
  • C. 6/7
  • D. 7/13
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 6, total outcomes = 13. So probability = 6/13.
Q14 Medium

A bag has 4 red and 2 blue balls. Probability of picking a red ball?

  • A. 2/6
  • B. 4/2
  • C. 6/4
  • D. 4/6 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 4, total outcomes = 6. So probability = 4/6.
Q15 Medium

From a 52-card deck, what is the probability of drawing a face card (J, Q, or K)?

  • A. 16/52
  • B. 13/52
  • C. 12/52 ✓
  • D. 10/52
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Each suit has J, Q, K so face cards = 3 x 4 = 12. Probability = 12/52 = 3/13.
Q16 Medium

A bag has 4 red and 8 blue balls. Probability of picking a red ball?

  • A. 12/4
  • B. 8/12
  • C. 4/8
  • D. 4/12 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 4, total outcomes = 12. So probability = 4/12.
Q17 Medium

From a standard 52-card deck, what is the probability of drawing a Diamond?

  • A. 1/4
  • B. 26/52
  • C. 13/52 ✓
  • D. 1/2
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. There are 13 diamonds in 52 cards. Probability = 13/52 = 1/4.
Q18 Medium

A bag has 6 red and 6 blue balls. Probability of picking a red ball?

  • A. 6/6
  • B. 12/6
  • C. 6/12 ✓
  • D. 6/12 (1)
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 6, total outcomes = 12. So probability = 6/12.
Q19 Medium

A bag has 1 red and 5 blue balls. Probability of picking a red ball?

  • A. 1/6 ✓
  • B. 1/5
  • C. 6/1
  • D. 5/6
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 1, total outcomes = 6. So probability = 1/6.
Q20 Medium

A bag has 5 red and 2 blue balls. Probability of picking a red ball?

  • A. 7/5
  • B. 2/7
  • C. 5/7 ✓
  • D. 5/2
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 7. So probability = 5/7.
Q21 Medium

A bag has 3 red and 4 blue balls. Probability of picking a red ball?

  • A. 7/3
  • B. 4/7
  • C. 3/4
  • D. 3/7 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 3, total outcomes = 7. So probability = 3/7.
Q22 Medium

From a 52-card deck, what is the probability of drawing a face card (J, Q, or K)?

  • A. 16/52
  • B. 12/52 ✓
  • C. 13/52
  • D. 10/52
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Each suit has J, Q, K so face cards = 3 x 4 = 12. Probability = 12/52 = 3/13.
Q23 Medium

A bag has 5 red and 6 blue balls. Probability of picking a red ball?

  • A. 11/5
  • B. 5/11 ✓
  • C. 5/6
  • D. 6/11
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 5, total outcomes = 11. So probability = 5/11.
Q24 Medium

A bag has 4 red and 2 blue balls. Probability of picking a red ball?

  • A. 4/2
  • B. 6/4
  • C. 2/6
  • D. 4/6 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 4, total outcomes = 6. So probability = 4/6.
Q25 Medium

A bag has 2 red and 3 blue balls. Probability of picking a red ball?

  • A. 2/5 ✓
  • B. 3/5
  • C. 2/3
  • D. 5/2
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 2, total outcomes = 5. So probability = 2/5.
Q26 Medium

From a standard 52-card deck, what is the probability of drawing a Diamond?

  • A. 1/4
  • B. 1/2
  • C. 13/52 ✓
  • D. 26/52
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. There are 13 diamonds in 52 cards. Probability = 13/52 = 1/4.
Q27 Medium

A bag has 6 red and 8 blue balls. Probability of picking a red ball?

  • A. 6/8
  • B. 8/14
  • C. 14/6
  • D. 6/14 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 6, total outcomes = 14. So probability = 6/14.
Q28 Medium

From a standard 52-card deck, what is the probability of drawing a Club?

  • A. 1/2
  • B. 13/52 ✓
  • C. 1/4
  • D. 26/52
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. There are 13 clubs in 52 cards. Probability = 13/52 = 1/4.
Q29 Medium

A bag has 3 red and 2 blue balls. Probability of picking a red ball?

  • A. 5/3
  • B. 3/2
  • C. 2/5
  • D. 3/5 ✓
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 3, total outcomes = 5. So probability = 3/5.
Q30 Medium

A bag has 3 red and 6 blue balls. Probability of picking a red ball?

  • A. 3/6
  • B. 3/9 ✓
  • C. 6/9
  • D. 9/3
Explanation: Probability = Number of favorable outcomes ÷ Total number of possible outcomes. For a standard deck: 52 cards, 4 suits (13 each). For dice: 6 outcomes per die. Favorable outcomes = 3, total outcomes = 9. So probability = 3/9.

Key Takeaways

  • Total outcomes for N coin tosses = 2^N. For N dice = 6^N.
  • Cards: 52 total = 4 suits × 13 ranks. Know the composition of the deck.
  • For 'or' events: P(A or B) = P(A) + P(B) - P(A and B).

Frequently Asked Questions

What is the basic formula for probability?

Probability = Number of favorable outcomes ÷ Total number of possible outcomes. Always express as a fraction, decimal, or percentage between 0 and 1.

What are independent and mutually exclusive events?

Independent: Outcome of one does NOT affect the other (e.g., rolling dice twice). P(A and B) = P(A) × P(B). Mutually exclusive: Events cannot occur together (e.g., getting heads and tails on one coin toss). P(A or B) = P(A) + P(B).

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