Chapter 14: Probability

Important questions — the ones the board asked most (20)

1. Q1Case-based4 marksAsked: 2026

   37. A group of friends wanted to play cards with two identical packs together. While shuffling the cards, three cards are dropped. Rest of the cards are shuffled and one card is drawn at random. Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, answer the following questions : (i) Find the probability that the drawn card is a face card. (ii) Find the probability that the drawn card is either a king or a queen. (iii) (a) Do you think that the probability of getting a queen was higher if none of the cards were dropped ? Justify your answer. OR (iii) (b) Find the probability that the drawn card is a jack. Compare it with the probability when none of the cards were dropped. In which case is the probability of getting a jack higher ?

   

   Medha Ma’am

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   Model answer · probability, playing cards

   1. 1.Two identical packs have 2 × 52 = 104 cards. Three cards are dropped: a queen of hearts, a ten of spades and an ace of clubs. Total number of cards = 2 × 52 – 3 = 101 The cards are shuffled, so all 101 outcomes are equally likely. (i) Kings, queens and jacks are called face cards. Two packs have 8 kings + 8 queens + 8 jacks = 24 face cards. One queen was dropped, so 23 face cards remain. P (a face card) = 23/1011 mark
   2. 2.(ii) The remaining cards include 8 kings and 8 – 1 = 7 queens, so 8 + 7 = 15 cards are favourable. P (either a king or a queen) = 15/1011 mark
   3. 3.(iii) (a) Yes, the probability of getting a queen was higher if none of the cards were dropped.0.5 marks
   4. 4.With no card dropped, there are 104 cards and 8 queens. P (a queen when no cards were dropped) = 8/1040.5 marks
   5. 5.With the three cards dropped, there are 101 cards and 7 queens. P (a queen when cards were dropped) = 7/1010.5 marks
   6. 6.To compare, cross-multiply: 8 × 101 = 808 and 7 × 104 = 728. ∵ 8/104 > 7/101 as 808 > 728 So probability of getting a queen was higher if none of the cards were dropped.0.5 marks

   • Begin by writing "Total number of cards = 2 × 52 – 3 = 101" before you find any probability.
   • Take away the dropped queen of hearts: count 23 face cards and 7 queens, not 24 and 8.
   • In (iii) (a), write "Yes", give both probabilities, 8/104 and 7/101, and then compare them using 808 > 728.

   Marks split as in the board’s published marking scheme

2. Q2Case-based4 marksAsked: 2026

   Some friends wanted to play cards with two identical decks of cards. While shuffling the cards, three cards fell down. They shuffled the remaining cards well and drew one card at random. Assuming that the three fallen cards are a queen of hearts, a ten of spades and an ace of clubs, answer the following questions : (i) What is the probability that the drawn card is a face card? (ii) Find the probability that the drawn card is a queen or a king. (iii) (a) Do you think that if no card had fallen, the probability of the drawn card being a queen would have been higher? Justify your answer. OR (iii) (b) Find the probability that the drawn card is a jack. Compare this with the probability obtained in the situation if no card had fallen. In which situation is the probability of the drawn card being a jack higher?

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 175 s

   Model answer · Probability, playing cards

   1. 1.Given: two identical decks of 52 cards each. Three cards fell down: a queen of hearts, a ten of spades and an ace of clubs. Total number of cards = 2 × 52 − 3 = 101 The cards were well-shuffled, so all 101 outcomes are equally likely. P(E) = number of outcomes favourable to E / number of all possible outcomes (i) Kings, queens and jacks are called face cards. Two decks have 2 × 12 = 24 face cards. The queen of hearts fell, so 24 − 1 = 23 face cards are left. P(a face card) = 23/1011 mark
   2. 2.(ii) Kings left = 8, because no king fell. Queens left = 8 − 1 = 7, because one queen fell. Number of outcomes favourable to drawing a queen or a king = 8 + 7 = 15 P(either a king or a queen) = 15/1011 mark
   3. 3.(iii) (a) Yes. If no card had fallen, the probability of drawing a queen would have been higher.0.5 marks
   4. 4.If no card had fallen, there would be 104 cards, and 8 of them would be queens. P(a queen when no cards were dropped) = 8/1040.5 marks
   5. 5.After the cards fell, 101 cards are left, and 7 of them are queens. P(a queen when cards were dropped) = 7/1010.5 marks
   6. 6.Cross-multiplying: 8 × 101 = 808 and 7 × 104 = 728. Since 808 > 728, 8/104 > 7/101. So the probability of getting a queen was higher if none of the cards were dropped.0.5 marks

   • Start by writing "Total number of cards = 2 × 52 − 3 = 101". Every probability uses this denominator.
   • Take away the fallen queen of hearts when you count face cards (23) and queens (7).
   • In (iii)(a), write "Yes", work out both 8/104 and 7/101, then compare them using 808 > 728.

   Marks split as in the board’s published marking scheme

3. Q3Case-based4 marksAsked: 2024

   In a survey of holidays, 120 persons were asked about the type of vehicle used in their last holiday. The following pie chart shows the results of this survey. On the basis of observation of the pie chart, answer the following questions: (i) If a person is chosen at random, find the probability of his/her travelling by bus or ship. (ii) Which is the most popular/favourite means of transport and how many people used it? (iii) (a) The probability of a randomly chosen person not travelling by train is 4/5, then find how many people used the train. or (iii) (b) The probability of a randomly chosen person travelling by aeroplane is 7/60, then find the income earned by the airline company, if they charged 5,000 per person.

4. Q4Case-based4 marksAsked: 2024

   Family structure : In a recent survey of this year, 51% of the families in the United States of America had no children, 20% had one child, 19% had two children, 7% had three children and 3% had four or more children. A family is selected at random. Based on the above information, answer the following questions : (i) Find the probability that the selected family has two or three children. (ii) Find the probability that the selected family has more than one child. (iii) (a) Find the probability that the selected family has less than three children. OR (b) Find the probability that the selected family has more than two children.

   How the model answer begins

   1. Given: out of every 100 families, 51 have no children, 20 have one child, 19 have two children, 7 have three children and 3 have four or more children. A family is selected at random, so all 100 outcomes are equally…

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5. Q5Case-based4 marksAsked: 2024

   Case Study - 3 Family structure : A recent survey found that for this year, in the United States 51% of families had no children, 20% of families had one child, 19% of families had two children, 7% of families had three children and 3% of families had four or more children. A family is chosen at random. Based on the above information, answer the following questions : (i) Find the probability of having two or three children in the chosen family. (ii) Find the probability of having more than one child in the chosen family. (iii) (a) Find the probability of having less than three children in the chosen family. OR (b) Find the probability of having more than two children in the chosen family.

   How the model answer begins

   1. Given: out of every 100 families, 51 have no children, 20 have one child, 19 have two children, 7 have three children and 3 have four or more children. A family is chosen at random, so there are 100 equally likely…

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6. Q6Case-based4 marksAsked: 2023

   A middle school decided to run the following spinner game as a fund-raiser on Christmas Carnival. Making Purple : Spin each spinner once. Blue and red make purple. So, if one spinner shows Red (R) and another Blue (B), then you 'win'. One such outcome is written as 'RB'. Based on the above, answer the following questions : (i) List all possible outcomes of the game. (ii) Find the probability of 'Making Purple'. (iii) (a) For each win, a participant gets 10, but if he/she loses, he/she has to pay 5 to the school. If 99 participants played, calculate how much fund could the school have collected. OR (iii) (b) If the same amount of 5 has been decided for winning or losing the game, then how much fund had been collected by school ? (Number of participants = 99)

7. Q7Case-based4 marksAsked: 2023

   Blood group describes the type of blood a person has. It is a classification of blood based on the presence or absence of inherited antigenic substances on the surface of red blood cells. Blood types predict whether a serious reaction will occur in a blood transfusion. In a sample of 50 people, 21 had type O blood, 22 had type A, 5 had type B and rest had type AB blood group. Based on the above, answer the following questions : (i) What is the probability that a person chosen at random had type O blood ? (ii) What is the probability that a person chosen at random had type AB blood group ? (iii) What is the probability that a person chosen at random had neither type A nor type B blood group ? OR (iii) What is the probability that person chosen at random had either type A or type B or type O blood group ?

   How the model answer begins

   1. (i) Given: the total number of people in the sample = 50. Choosing a person at random means every person is equally likely to be chosen, so the number of possible outcomes = 50. Let O be the event 'the person has type O…

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8. Q8Case-based4 marksAsked: 2025

   Rahul is a lucky charm for his cricket team. He has a jar of cards with numbers from 10 to 74. Before each match, he draws a card from the jar. If the card bears an even number, the team wins. If the number is even and divisible by 5, they win by a big margin. If the number is an odd number less than 30, they win by a small margin. And if the number is a prime number between 50 and 74, they lose. Answer the following questions if Rahul draws a card today : (i) What is the probability that Rahul draws a card with an even number ? 1 (ii) What is the probability that Rahul draws a card with an odd number less than 30 ? 1 (iii) (a) What is the probability that Rahul draws a card with a prime number between 50 and 74 ? 2 OR (b) What is the probability that Rahul draws a card with an even number divisible by 5 ? 2

9. Q9Case-based4 marksAsked: 2023

   Computer-based learning (CBL) refers to any teaching methodology that makes use of computers for information transmission. At an elementary school level, computer applications can be used to display multimedia lesson plans. A survey was done on 1000 elementary and secondary schools of Assam and they were classified by the number of computers they had. | Number of Computers | 1 – 10 | 11 – 20 | 21 – 50 | 51 – 100 | 101 and more | | Number of Schools | 250 | 200 | 290 | 180 | 80 | One school is chosen at random. Then : (i) Find the probability that the school chosen at random has more than 100 computers. (ii) (a) Find the probability that the school chosen at random has 50 or fewer computers. OR (ii) (b) Find the probability that the school chosen at random has no more than 20 computers. (iii) Find the probability that the school chosen at random has 10 or less than 10 computers.

   How the model answer begins

   1. Total schools = 250 + 200 + 290 + 180 + 80 = 1000. Schools with more than 100 computers (101 and more) = 80. P = 80/1000 = 2/25.

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10. Q10Case-based4 marksAsked: 2023

    [passage] Number of Computers | 1 – 10 | 11 – 20 | 21 – 50 | 51 – 100 | 101 and more Number of Schools | 250 | 200 | 290 | 180 | 80 One school is chosen at random. Then : (i) Find the probability that the school chosen at random has more than 100 computers. 1 (ii) (a) Find the probability that the school chosen at random has 50 or fewer computers. 2 OR (ii) (b) Find the probability that the school chosen at random has no more than 20 computers. 2 (iii) Find the probability that the school chosen at random has 10 or less than 10 computers. 1

11. Q11Case-based4 marksAsked: 2023

    Computer-based learning (CBL) refers to any teaching methodology that makes use of computers for information transmission. At an elementary school level, computer applications can be used to display multimedia lesson plans. A survey was done on 1000 elementary and secondary schools of Assam and they were classified by the number of computers they had. Number of Computers: 1-10, 11-20, 21-50, 51-100, 101 and more Number of Schools: 250, 200, 290, 180, 80 One school is chosen at random. Then : (i) Find the probability that the school chosen at random has more than 100 computers. (ii) (a) Find the probability that the school chosen at random has 50 or fewer computers. OR (ii) (b) Find the probability that the school chosen at random has no more than 20 computers. (iii) Find the probability that the school chosen at random has 10 or less than 10 computers.

12. Q12Case-based4 marksAsked: 2023

    "Eight Ball" is a game played on a pool table with 15 balls numbered 1 to 15 and a "cue ball" that is solid and white. Of the 15 numbered balls, eight are solid (non-white) coloured and numbered 1 to 8 and seven are striped balls numbered 9 to 15. The 15 numbered pool balls (no cue ball) are placed in a large bowl and mixed, then one ball is drawn out at random. Based on the above information, answer the following questions : (i) What is the probability that the drawn ball bears number 8 ? (ii) What is the probability that the drawn ball bears an even number ? OR What is the probability that the drawn ball bears a number, which is a multiple of 3 ? (iii) What is the probability that the drawn ball is a solid coloured and bears an even number ?

13. Q13Case-based4 marksAsked: 2023

    Case Study - 3 Some students were asked to list their favourite colour. The measure of each colour is shown by the central angle of a pie chart given below : Study the pie chart and answer the following questions : (i) If a student is chosen at random, then find the probability of his/her favourite colour being white ? (ii) What is the probability of his/her favourite colour being blue or green ? (iii) (a) If 15 students liked the colour yellow, how many students participated in the survey ? OR (iii) (b) What is the probability of the favourite colour being red or blue ?

    How the model answer begins

    1. The central angle for white is 120°. P(white) = 120°/360° = 1/3.

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14. Q14Case-based4 marksAsked: 2023

    Some students were asked to list their favourite colour. The measure of each colour is shown by the central angle of a pie chart given below : [passage] Study the pie chart and answer the following questions : (i) If a student is chosen at random, then find the probability of his/her favourite colour being white ? 1 (ii) What is the probability of his/her favourite colour being blue or green ? 1 (iii) (a) If 15 students liked the colour yellow, how many students participated in the survey ? 2 OR (iii) (b) What is the probability of the favourite colour being red or blue ? 2

15. Q15Short answer3 marksAsked: 2026

    31. Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that : (i) same number appears on both the dice ? (ii) different number appears on both the dice ?

    How the model answer begins

    1. When two dice of different colours are thrown at the same time, each die can show any one of the numbers 1, 2, 3, 4, 5, 6. In each ordered pair, the first number is the number on the first die and the second number is…

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16. Q16Short answer3 marksAsked: 2026

    Two different coins are tossed simultaneously. What is the probability of getting : (i) at least one head ? (ii) at most one tail ? (iii) a head and a tail ?

    How the model answer begins

    1. When two different coins are tossed simultaneously, the possible outcomes are HH, HT, TH, TT, which are all equally likely. Number of all possible outcomes = 4 P(E) = Number of outcomes favourable to E / Number of all…

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17. Q17Short answer3 marksAsked: 2026

    Two dice are thrown at the same time. Determine the probability that (i) sum of the numbers on the two dice is 5, and (ii) difference of the numbers on the two dice is 3.

    How the model answer begins

    1. When two dice are thrown at the same time, each die can show any one of the numbers 1, 2, 3, 4, 5, 6. Each outcome is an ordered pair (number on first die, number on second die), so (1, 4) is different from (4, 1).…

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18. Q18Short answer3 marksAsked: 2026

    Two different dice are thrown together. Find the probability that the numbers obtained have : (i) even sum, (ii) even product.

    How the model answer begins

    1. Given: two different dice are thrown together. Each die can show 1, 2, 3, 4, 5 or 6. Number of all possible outcomes = 6 × 6 = 36, and these outcomes are equally likely. (i) Let E be the event 'the sum is even'. The sum…

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19. Q19Short answer3 marksAsked: 2026

    Two dice are thrown together. Find the probability of the following: (i) The sum of the numbers obtained is 10. (ii) The product of the numbers obtained is 6.

    How the model answer begins

    1. When two dice are thrown together, each outcome is an ordered pair (number on the first die, number on the second die). Each die can show any of 1 to 6. Number of possible outcomes = 6 × 6 = 36 (i) Let E be the event…

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20. Q20Short answer3 marksAsked: 2026

    Two dice are rolled together. Find the probability that at least one of the numbers obtained is a multiple of 3.

    How the model answer begins

    1. Given: two dice are rolled together. Each die can show any one of the numbers 1, 2, 3, 4, 5, 6, and an outcome is written as an ordered pair (number on first die, number on second die). All outcomes are equally likely.…

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46 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   A die is thrown once. Probability of getting a number other than 3 is :

   • (A) 1/6
   • (B) 3/6
   • (C) 5/6 ✓
   • (D) 1

   Answer: (C)

2. 2. CBSE 2026

   Assertion (A) : The probability that a leap year has 53 Mondays is 2/7. Reason (R) : The probability that a non-leap year has 53 Mondays is 5/7.

   • (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
   • (B) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false. ✓
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (C)

3. 3. CBSE 2026

   For an event E, P(E) + P(E_bar) = x, then the value of x^2 - 3 is :

   • (A) -2 ✓
   • (B) 2
   • (C) 1
   • (D) -1

   Answer: (A)

4. 4. CBSE 2026

   The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is :

   • (A) 15/25 ✓
   • (B) 10/25
   • (C) 11/25
   • (D) 9/25

   Answer: (A)

5. 5. CBSE 2026

   Two dice are rolled together. The probability that the sum of the numbers obtained is divisible by 6, is :

   • (A) 1/6 ✓
   • (B) 11/36
   • (C) 1/12
   • (D) 1/4

   Answer: (A)

6. 6. CBSE 2026

   Three coins are tossed together. The probability of getting exactly one head, is :

   • (A) 1/8
   • (B) 3/4
   • (C) 1/2
   • (D) 3/8 ✓

   Answer: (D)

7. 7. CBSE 2026

   Probability of getting an irrational number at random from the numbers sqrt(3), sqrt(4), 3sqrt(9), 3sqrt(8), sqrt(5), 0, 4^(3/2) is :

   • (A) 0
   • (B) 4/7
   • (C) 3/7 ✓
   • (D) 1

   Answer: (C)

8. 8. CBSE 2026

   A card is drawn from a well shuffled deck of 52 cards. The probability of getting an ace or a ten is :

   • (A) 1/13
   • (B) 2/13 ✓
   • (C) 1/26
   • (D) 4/13

   Answer: (B)

9. 9. CBSE 2026

   (Assertion - Reason based questions) Direction : In question numbers 19 and 20, an Assertion (A) is followed by a Reason (R). Choose the correct option : Assertion (A) : The probability of occurrence of an event cannot be 1/0.9. Reason (R) : For any event (E), 0 <= P(E) <= 1.

   • (A) Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A). ✓
   • (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false.
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (A)

10. 10. CBSE 2026

    The probability of choosing an irrational number at random from the numbers sqrt(3), sqrt(4), 3sqrt(9), 3sqrt(8), sqrt(5), 0, 4^(3/2) is:

    • (A) 0
    • (B) 4/7
    • (C) 3/7 ✓
    • (D) 1

    Answer: (C)

Questions parents and students ask

How many marks does "Probability" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 5 marks and about 3 questions a year; by marks it ranks 11 in Maths.

Which are the most important (VVI) questions of "Probability"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 66 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.