Class 10 Maths Test Paper Chapter 14 Probability | CBSE Board | Detailed Solutions & Marking Scheme
VEDANT CLASSES – A Pathway to Success…
Class: 10
Subject: Mathematics
Chapter: Probability (Coins, Dice, Cards, At least/At most, Leap Year)
Board: CBSE
Marks: 30
Section A (1 × 7 = 7 Marks)
(5 MCQs + 2 Assertion Reason)
Q1. A game involves tossing a fair coin. A student wins if head appears. What is the probability of winning?
(A) 0 (B) 1/2 (C) 1 (D) 1/4
Q2. Two dice are thrown in a game. What is the probability that the difference of numbers is 3?
(A) 1/9 (B) 1/6 (C) 1/12 (D) 1/18
Q3. A card is drawn from a standard deck of 52 cards. What is the probability of getting a card which is not an ace?
(A) 1/13 (B) 4/13 (C) 12/13 (D) 9/13
Q4. A student selects a month randomly. What is the probability that it has 31 days?
(A) 7/12 (B) 5/12 (C) 1/2 (D) 1/3
Q5. Two coins are tossed. What is the probability of getting at least one head?
(A) 1/4 (B) 1/2 (C) 3/4 (D) 1
Assertion Reason Questions
Q6.
Assertion (A): The probability of getting a tail when a fair coin is tossed is 1/2.
Reason (R): Total outcomes are always equal to favourable outcomes.
(A) Both A and R are true and R is correct explanation
(B) Both A and R are true but R is not correct explanation
(C) A is true but R is false
(D) A is false but R is true
Q7.
Assertion (A): Probability of an event lies between 0 and 1.
Reason (R): Probability can never be negative or greater than 1.
(A) Both A and R are true and R is correct explanation
(B) Both A and R are true but R is not correct explanation
(C) A is true but R is false
(D) A is false but R is true
Section B (2 × 4 = 8 Marks)
Q8.
A bag contains 4 red and 6 blue balls. A ball is drawn randomly. Find the probability that the ball is not blue.
Q9.
A die is thrown once in a board game. Find the probability of getting:
(i) a multiple of 2
(ii) a number greater than 4
Q10.
From numbers 1 to 20, a number is selected randomly. Find the probability that it is:
(i) divisible by 5
(ii) a prime number
Q11.
A student randomly selects a day of a leap year. Find the probability that it is:
(i) a Monday (assume all days equally likely)
(ii) not a Sunday
Section C (3 × 2 = 6 Marks)
Q12.
Two coins are tossed simultaneously. Find the probability of:
(i) exactly one head
(ii) at most one tail
(iii) no head
Q13.
A card is drawn from a well-shuffled deck. Find the probability that the card is:
(i) a red card
(ii) a king or a queen
(iii) not a face card
Section D (5 × 1 = 5 Marks)
Q14. (Case-Based Game Situation)
In a school carnival game, a player throws two dice simultaneously.
Find the probability that:
(i) sum is 8
(ii) sum is at most 5
(iii) sum is a multiple of 3
(iv) sum is a prime number
(v) sum is greater than 10
Section E (Case Study – 4 Marks)
Q15. (Cricket Match Scenario)
In a cricket match, a spinner decides to toss a coin before each over. If head comes, he bowls spin; if tail comes, fast bowling is chosen. This is done for 2 overs.
Answer the following:
(i) What is the probability that both overs have spin bowling? (1 mark)
(ii) What is the probability of getting exactly one fast bowling? (1 mark)
(iii) What is the probability of getting at least one spin bowling? (2 marks)
🔥 Paper Highlights (CBSE Trend Based)
Dice difference = classic PYQ
Cards (not ace, face cards) = frequently asked
“At least / At most” = very important concept
Leap year/day questions = recent CBSE pattern
Here is the DETAILED SOLUTION + MARKING SCHEME + PAPER ANALYSIS + PRO TIPS
(Kept in plain, easy-to-copy format for your students)
✅ SECTION A SOLUTIONS (1 mark each)
Q1.
Total outcomes = {H, T} = 2
Favourable (Head) = 1
Probability = 1/2
Answer: (B) ✅
Marks: 1
Q2.
Total outcomes when two dice thrown = 36
Pairs with difference = 3 → (1,4), (4,1), (2,5), (5,2), (3,6), (6,3)
Favourable = 6
Probability = 6/36 = 1/6
Answer: (B) ✅
Marks: 1
Q3.
Total cards = 52
Aces = 4
Not ace = 52 − 4 = 48
Probability = 48/52 = 12/13
Answer: (C) ✅
Marks: 1
Q4.
Months with 31 days = 7
Probability = 7/12
Answer: (A) ✅
Marks: 1
Q5.
Outcomes = HH, HT, TH, TT
At least one head = HH, HT, TH → 3 outcomes
Probability = 3/4
Answer: (C) ✅
Marks: 1
Q6.
A is TRUE
R is FALSE (total ≠ favourable always)
Answer: (C) ✅
Marks: 1
Q7.
Both true and R explains A
Answer: (A) ✅
Marks: 1
✅ SECTION B SOLUTIONS (2 marks each)
Q8.
Total balls = 4 + 6 = 10
Not blue = red = 4
Probability = 4/10 = 2/5
Marking:
Formula (1) + Calculation (1) = 2 marks
Q9.
Total outcomes = 6
(i) Multiples of 2 → {2,4,6} = 3
P = 3/6 = 1/2
(ii) Numbers > 4 → {5,6} = 2
P = 2/6 = 1/3
Marking:
Each part 1 mark → 2 marks
Q10.
Total numbers = 20
(i) Divisible by 5 → {5,10,15,20} = 4
P = 4/20 = 1/5
(ii) Prime numbers → {2,3,5,7,11,13,17,19} = 8
P = 8/20 = 2/5
Marks: 1 + 1 = 2 marks
Q11.
Leap year = 366 days
(i) Monday → 366/7 = approx equal → 1/7
(ii) Not Sunday → 6/7
Marks: 1 + 1 = 2 marks
✅ SECTION C SOLUTIONS (3 marks each)
Q12.
Outcomes = HH, HT, TH, TT
(i) Exactly one head → HT, TH = 2
P = 2/4 = 1/2
(ii) At most one tail → HH, HT, TH = 3
P = 3/4
(iii) No head → TT = 1
P = 1/4
Marking:
Each part 1 mark → 3 marks
Q13.
Total cards = 52
(i) Red cards = 26
P = 26/52 = 1/2
(ii) King or Queen = 4 + 4 = 8
P = 8/52 = 2/13
(iii) Face cards = 12
Not face = 52 − 12 = 40
P = 40/52 = 10/13
Marks: 1 + 1 + 1 = 3 marks
✅ SECTION D SOLUTION (5 marks)
Q14.
Total outcomes = 36
(i) Sum = 8 → (2,6),(6,2),(3,5),(5,3),(4,4) = 5
P = 5/36
(ii) Sum ≤ 5 → (1,1),(1,2),(2,1),(1,3),(3,1),(2,2),(1,4),(4,1),(2,3),(3,2) = 10
P = 10/36 = 5/18
(iii) Multiple of 3 → sums 3,6,9,12 → favourable = 12
P = 12/36 = 1/3
(iv) Prime sums → 2,3,5,7,11 → favourable = 15
P = 15/36 = 5/12
(v) Sum > 10 → 11,12 → favourable = 3
P = 3/36 = 1/12
Marking:
Each part 1 mark → 5 marks
✅ SECTION E SOLUTION (Case Study – 4 marks)
Outcomes = HH, HT, TH, TT
(i) Both spin (HH)
P = 1/4
Marks: 1
(ii) Exactly one fast (HT, TH)
P = 2/4 = 1/2
Marks: 1
(iii) At least one spin (HH, HT, TH)
P = 3/4
Marks:
Concept (1) + Answer (1) = 2 marks
📊 PAPER ANALYSIS (VERY IMPORTANT)
🔥 Difficulty Level
Section A → Easy
Section B → Easy to Moderate
Section C → Moderate
Section D → Moderate (calculation heavy)
Section E → Easy
👉 Overall: CBSE Board Level (Moderate)
📌 Chapter Weightage Insight
Dice → 35% paper
Cards → 20%
Coins → 20%
At least/At most → 15%
Leap year → 10%
⚠️ Common Student Mistakes
Writing wrong sample space (36 mistakes)
Confusion in at least / at most
Missing outcomes in dice sums
Not simplifying fractions
Mixing face cards count (12 vs 16 mistake)
🎯 PRO TIPS FOR BOARD EXAM
✅ 1. Always write formula
Probability = Favourable outcomes / Total outcomes
👉 Even in MCQ → helps avoid silly mistakes
✅ 2. Dice questions = MAKE TABLE mentally
36 outcomes fixed
Memorize common sums:
Sum = 7 → 6 ways
Sum = 8 → 5 ways
✅ 3. Cards = Learn fixed data
Total = 52
Red = 26
Face = 12
Ace = 4
✅ 4. “At least” trick
At least one = 1 − (none)
Example:
At least one head = 1 − P(no head)
✅ 5. Time Strategy
Section A → 5 min
Section B → 8 min
Section C → 10 min
Section D → 10 min
Section E → 7 min
🚀 Topper Strategy (Game Changer)
👉 Solve Section D first (5 marks secured fast)
👉 Then easy MCQs
👉 Leave tricky “at most” for last
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