Chapter 14: Probability
CBSE Official PYQs (2015-2026)
Q1. If a card is drawn from a well-shuffled deck of 52 playing cards, then the probability of getting a black face card is :
Q2. In a group of 20 people, 5 cannot swim. If one person is selected at random, then the probability that he/she can swim is :
Q3. If for any event \(E\), \(P(E) + P(\bar{E}) = q\), then the value of \(q^2 - 3\) is :
Q4. Which of the following cannot be the probability of an event?
Q5. From the numbers 1, 2, 3, ..., 25, a number is chosen at random. The probability of it being a composite number is :
Q6. The probability of getting a bad egg in a lot of 400 eggs is 0.035. The number of bad eggs in the lot is :
Q7. Two dice are thrown together. The probability of getting the same number on both dice (a doublet) is :
Q8. A letter of English alphabets is chosen at random. What is the probability that it is a letter of the word 'MATHEMATICS'?
Q9. If a card is drawn at random from a well-shuffled deck of 52 playing cards, then the probability of getting a red king is :
Q10. A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets are sold, how many tickets has she bought?
Q11. A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. The probability that the selected ticket has a number which is a multiple of 5 is :
Q12. In a leap year, the probability of having 53 Sundays is :
Q13. A bag contains 3 red, 5 black and 7 white balls. A ball is drawn from the bag at random. The probability that the ball drawn is not black is :
Q14. If a die is thrown once, the probability of getting a prime number is :
Q15. An integer is chosen between 70 and 100. What is the probability that it is a prime number?
Q16. Someone is asked to take a number from 1 to 100. The probability that it is a prime number is :
Q17. Two coins are tossed simultaneously. What is the probability of getting at most one head?
Q18. A card is drawn from a deck of 52 cards. The event \(E\) is that the card is not an ace of hearts. The number of outcomes favourable to \(E\) is :
Q19. A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime number less than 23 is :
Q20. If a card is drawn from a well-shuffled deck of 52 playing cards, then the probability of getting a non-face card is :
Q21. Which of the following cannot be the probability of an event?
Q22. The probability of happening of an event is \(p\). The probability of non-happening of this event is :
Q23. If two different dice are rolled together, then the probability that the sum of the two numbers appearing on top of the dice is 9 is :
Q24. In a box there are 8 red, 7 blue and 6 green marbles. One marble is picked up at random. What is the probability that it is neither red nor green?
Q25. A card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card drawn is a spade or an ace?
Q26. If the probability of winning a game is 0.4, then the probability of losing it is :
Q27. A box contains cards numbered 6 to 50. A card is drawn at random from the box. The probability that the drawn card has a number which is a perfect square is :
Q28. Three coins are tossed simultaneously. What is the probability of getting exactly two tails?
Q29. The probability of getting a non-leap year having 53 Mondays is :
Q30. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. What is the probability that it will point at an odd number?
Q31. Assertion (A): The probability of getting a prime number when a die is thrown once is 1/2.
Reason (R): Prime numbers on a die are 2, 3, 5.
Q32. Assertion (A): If a coin is tossed twice, the probability of getting at least one tail is 3/4.
Reason (R): The sample space when a coin is tossed twice is {H, T, HT, TH}.
Q33. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.
Q34. Two different dice are rolled together. Find the probability that:
(i) the sum of the numbers on the two dice is more than 9.
(ii) the product of the numbers on the two dice is a perfect square.
(i) \(\frac{1}{6}\) (Outcomes with sum > 9: (4,6), (5,5), (5,6), (6,4), (6,5), (6,6) \(\implies 6/36 = 1/6\))
(ii) \(\frac{2}{9}\) (Favourable outcomes: (1,1), (1,4), (2,2), (3,3), (4,1), (4,4), (5,5), (6,6) \(\implies 8/36 = 2/9\))
Q35. A game consists of tossing a one-rupee coin three times and noting its outcome each time. Hanif wins if all the tosses give the same result, i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
Q36. A box contains 80 cards numbered from 1 to 80. A card is drawn at random from the box. Find the probability that it bears:
(i) a two-digit number.
(ii) a perfect square number.
(i) \(\frac{71}{80}\) (Two-digit numbers: 10 to 80 [71 numbers])
(ii) \(\frac{1}{10}\) (Perfect squares: {1, 4, 9, 16, 25, 36, 49, 64} [8 numbers] \(\implies 8/80 = 1/10\))
Q37. Two coins are tossed simultaneously. Find the probability of getting:
(i) exactly one tail.
(ii) no head.
(i) \(\frac{1}{2}\) ({HT, TH} \(\implies 2/4 = 1/2\))
(ii) \(\frac{1}{4}\) ({TT} \(\implies 1/4\))
Q38. From a pack of 52 playing cards, jacks, queens and kings of red colour are removed. From the remaining a card is drawn at random. Find the probability that the drawn card is:
(i) a black face card.
(ii) a red card.
(i) \(\frac{3}{23}\) (Remaining cards = \(52 - 6 = 46\); Black face cards = 6 \(\implies 6/46 = 3/23\))
(ii) \(\frac{10}{23}\) (Remaining red cards = \(26 - 6 = 20 \implies 20/46 = 10/23\))
Q39. A die is thrown twice. What is the probability that:
(i) 5 will not come up either time?
(ii) 5 will come up at least once?
(i) \(\frac{25}{36}\) (Outcomes with no 5 = 25 \(\implies 25/36\))
(ii) \(\frac{11}{36}\) (Outcomes with at least one 5 = 11 \(\implies 11/36\))
Q40. A bag contains cards numbered from 11 to 60. A card is drawn at random from the bag. Find the probability that the number on the drawn card is:
(i) an odd number.
(ii) a perfect square number.
(i) \(\frac{1}{2}\) (Total cards = 50; Odd numbers = 25 \(\implies 25/50 = 1/2\))
(ii) \(\frac{2}{25}\) (Perfect squares: {16, 25, 36, 49} [4 numbers] \(\implies 4/50 = 2/25\))
Q41. An integer is chosen at random from 1 to 100. Find the probability that the chosen integer is:
(i) divisible by 8.
(ii) not divisible by 8.
(i) \(\frac{3}{25}\) (Multiples of 8: {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96} [12 numbers] \(\implies 12/100 = 3/25\))
(ii) \(\frac{22}{25}\) (\(1 - 3/25 = 22/25\))
Q42. All red face cards are removed from a pack of 52 playing cards. A card is then drawn at random from the remaining pack. Find the probability that the drawn card is:
(i) a face card.
(ii) a card of clubs.
(iii) a red card.
(i) \(\frac{3}{23}\) (Remaining cards = 46; Remaining face cards = 6 black face cards \(\implies 6/46 = 3/23\))
(ii) \(\frac{13}{46}\) (Club cards are untouched \(\implies 13/46\))
(iii) \(\frac{10}{23}\) (Remaining red cards = 20 \(\implies 20/46 = 10/23\))
Q43. Two different dice are tossed together. Find the probability that:
(i) the sum of the two numbers is even.
(ii) the product of the two numbers is odd.
(iii) the sum of the two numbers is a multiple of 3.
(i) \(\frac{1}{2}\) (18 outcomes out of 36 \(\implies 1/2\))
(ii) \(\frac{1}{4}\) (Odd products only from odd \(\times\) odd: \(3 \times 3 = 9\) outcomes \(\implies 9/36 = 1/4\))
(iii) \(\frac{1}{3}\) (Sums of 3, 6, 9, 12 [12 outcomes] \(\implies 12/36 = 1/3\))
Q44. A bag contains 18 balls out of which \(x\) balls are red.
(i) If one ball is drawn at random from the bag, what is the probability that it is a red ball?
(ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be 9/8 times that of the previous probability. Find the value of \(x\).
(i) \(\frac{x}{18}\)
(ii) \(x = 8\) (\(\frac{x+2}{20} = \frac{9}{8} \cdot \frac{x}{18} \implies \frac{x+2}{20} = \frac{x}{16} \implies 16x + 32 = 20x \implies 4x = 32 \implies x = 8\))
Q45. A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability of getting:
(i) a king of a black colour.
(ii) a spade.
(iii) a face card.
(i) \(\frac{1}{26}\) (2 black kings / 52 = 1/26)
(ii) \(\frac{1}{4}\) (13 spades / 52 = 1/4)
(iii) \(\frac{3}{13}\) (12 face cards / 52 = 3/13)
Q46. Case Study: Board Game Tournament
In a school board game tournament, students are playing a game that uses a customized octagonal spinner and a standard 6-sided die. The octagonal spinner has faces numbered 1, 2, 3, 4, 5, 6, 7, and 8.
Based on the above information, answer the following questions:
- What is the probability that the spinner stops at a prime number? (1 Mark)
- If a player spins the octagonal spinner and rolls the 6-sided die simultaneously, find the probability that both show the number 5. (1 Mark)
- (a) Find the probability that the sum of the number indicated on the spinner and the number rolled on the die is at least 12. (2 Marks)
OR
(b) Find the probability that the product of the number on the spinner and the number on the die is a perfect square. (2 Marks)
1. \(\frac{1}{2}\) (Primes on spinner: {2, 3, 5, 7} [4 numbers] \(\implies 4/8 = 1/2\))
2. \(\frac{1}{48}\) (\(\frac{1}{8} \times \frac{1}{6} = \frac{1}{48}\))
3. (a) \(\frac{1}{8}\) (Sums \(\ge 12\): (6,6), (7,5), (7,6), (8,4), (8,5), (8,6) [6 outcomes] \(\implies 6/48 = 1/8\))
OR
3. (b) \(\frac{1}{6}\) (Square products: (1,1), (4,1), (1,4), (4,4), (2,2), (8,2), (3,3), (6,6) [8 outcomes] \(\implies 8/48 = 1/6\))
Q47. Case Study: Quality Control in Manufacturing
A factory manufactures light bulbs. A quality control inspector randomly selects and tests batches of 500 bulbs. In a particular batch, the probability of finding a defective bulb is 0.012.
Based on the above information, answer the following questions:
- Find the expected number of defective bulbs in this batch of 500. (1 Mark)
- If the inspector picks one bulb at random, what is the probability that it is non-defective? (1 Mark)
- (a) If 5 more defective bulbs are accidentally added to the batch of 500 before testing, find the new probability of picking a defective bulb. (2 Marks)
OR
(b) If 100 non-defective bulbs are added to the batch of 500, find the new probability of picking a non-defective bulb. (2 Marks)
1. 6 defective bulbs (\(500 \times 0.012 = 6\))
2. 0.988 (\(1 - 0.012 = 0.988\))
3. (a) \(\frac{11}{505}\) (New defective = 11, Total = 505 \(\implies 11/505\))
OR
3. (b) \(\frac{99}{100}\) (New non-defective = \(494 + 100 = 594\), Total = 600 \(\implies 594/600 = 99/100 = 0.99\))
Q48. Case Study: Card Game Challenge
In a card game, players select cards from a special custom deck containing 40 cards numbered 1 to 40. A player wins a bonus point if they draw a card matching a specific condition.
Based on the above information, answer the following questions:
- Find the probability that a player draws a card bearing a number which is a multiple of both 3 and 5. (1 Mark)
- Find the probability that the selected card bears a number which is a perfect cube. (1 Mark)
- (a) Find the probability that the card bears a two-digit prime number. (2 Marks)
OR
(b) Find the probability that the card bears a number which is divisible by 7 but not by 2. (2 Marks)
1. \(\frac{1}{20}\) (Multiples of 15: {15, 30} [2 numbers] \(\implies 2/40 = 1/20\))
2. \(\frac{3}{40}\) (Perfect cubes: {1, 8, 27} [3 numbers] \(\implies 3/40\))
3. (a) \(\frac{1}{5}\) (Two-digit primes: {11, 13, 17, 19, 23, 29, 31, 37} [8 numbers] \(\implies 8/40 = 1/5\))
OR
3. (b) \(\frac{3}{40}\) (Divisible by 7 and odd: {7, 21, 35} [3 numbers] \(\implies 3/40\))