USTET Mathematics — Statistics & ProbabilityExam Answer Templates

Marks

2

Topic

Measures of Central Tendency

Difficulty

easy

Template Id

T1

Examiner Tip

Always show the sum calculation explicitly - don't just write the final division

Model Answer

Given: Data set = {12, 15, 18, 22, 25, 30} To Find: Mean Solution: Mean = Sum of all observations / Number of observations Mean = (12 + 15 + 18 + 22 + 25 + 30) / 6 Mean = 122 / 6 Mean = 20.33 Therefore, the mean is 20.33

Question Type

short_answer

Answer Structure

• Line 1: Write Given and To Find [0.5 marks]
• Line 2: State the formula for mean [0.5 marks]
• Line 3: Substitute values and calculate sum [0.5 marks]
• Line 4: Final division and answer [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Not showing the addition step
• Calculation errors
• Missing units or final statement

Key Phrases To Include

• Mean = Sum/Number
• Given
• Therefore

Marks

3

Topic

Measures of Central Tendency

Difficulty

medium

Template Id

T2

Examiner Tip

Always verify your answer by checking if the median condition is satisfied

Model Answer

Given: Five numbers with median = 15 Four known numbers: 8, 12, 18, 20 To Find: Fifth number Solution: Let the fifth number be x. For median to be 15, when arranged in order, the middle (3rd) number must be 15. Case 1: If x ≤ 15 Arranged order: 8, 12, x, 15, 18, 20 (not possible since we need 15 as median) Case 2: If x > 15 Arranged order: 8, 12, 15, x, 18, 20 (not possible since 15 is not in our original set) Therefore, x = 15 Verification: Arranged order: 8, 12, 15, 18, 20 Median = 15 ✓ Therefore, the fifth number is 15.

Question Type

short_answer

Answer Structure

• Line 1: State given information clearly [0.5 marks]
• Line 2: Define variable for unknown [0.5 marks]
• Line 3: Apply median concept - middle value [1 mark]
• Line 4: Logical reasoning about position [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not considering different cases
• Not verifying the answer
• Incorrect understanding of median

Key Phrases To Include

• Let the fifth number be
• middle value
• arranged in order
• verification

Marks

3

Topic

Basic Probability

Difficulty

medium

Template Id

T3

Examiner Tip

Always list the sample space systematically - use a tree diagram if needed

Model Answer

Given: Fair coin tossed 3 times To Find: P(exactly 2 heads) Solution: Sample space for 3 tosses: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} n(S) = 8 Favorable outcomes (exactly 2 heads): E = {HHT, HTH, THH} n(E) = 3 P(exactly 2 heads) = n(E)/n(S) = 3/8 Therefore, the probability of getting exactly 2 heads is 3/8 or 0.375

Question Type

short_answer

Answer Structure

• Line 1: Write given and to find [0.5 marks]
• Line 2: List complete sample space [1 mark]
• Line 3: Identify favorable outcomes [1 mark]
• Line 4: Apply probability formula [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Incomplete sample space
• Missing favorable outcomes
• Not simplifying fraction

Key Phrases To Include

• Sample space
• favorable outcomes
• P(E) = n(E)/n(S)

Marks

5

Topic

Measures of Dispersion

Difficulty

hard

Template Id

T4

Examiner Tip

Show all calculations in a tabular format for clarity and to avoid arithmetic errors

Model Answer

Given: Data set = {2, 4, 6, 8, 10} To Find: Standard deviation (σ) Solution: Step 1: Calculate the mean Mean (x̄) = (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6 Step 2: Calculate deviations from mean (xi - x̄): (2-6), (4-6), (6-6), (8-6), (10-6) Deviations: -4, -2, 0, 2, 4 Step 3: Calculate squared deviations (xi - x̄)²: (-4)², (-2)², (0)², (2)², (4)² Squared deviations: 16, 4, 0, 4, 16 Step 4: Calculate variance Variance (σ²) = Σ(xi - x̄)²/n = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8 Step 5: Calculate standard deviation Standard deviation (σ) = √variance = √8 = 2√2 = 2.828 Therefore, the standard deviation is 2.828 (or 2√2).

Question Type

long_answer

Answer Structure

• Step 1: Calculate mean [1 mark]
• Step 2: Find deviations from mean [1 mark]
• Step 3: Calculate squared deviations [1 mark]
• Step 4: Calculate variance [1 mark]
• Step 5: Find standard deviation [1 mark]

Scoring Breakdown

Common Mark Deductions

• Calculation errors in any step
• Not showing squared deviations clearly
• Forgetting to take square root for final answer

Key Phrases To Include

• deviations from mean
• squared deviations
• variance
• standard deviation = √variance

Marks

1

Topic

Measures of Central Tendency

Difficulty

easy

Template Id

T5

Examiner Tip

Count frequencies systematically and state which value appears most often

Model Answer

Mode = 7 (appears 3 times, most frequent)

Question Type

very_short_answer

Answer Structure

• State the mode value and frequency [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not identifying the correct frequency
• Giving multiple values without explanation

Key Phrases To Include

• most frequent
• appears

Marks

3

Topic

Probability with Sets

Difficulty

medium

Template Id

T6

Examiner Tip

Draw a Venn diagram to visualize the problem - it helps avoid double counting

Model Answer

Given: Total students = 30 Like Math = 18, Like Physics = 12 Like both = 8 To Find: P(likes neither subject) Solution: Using the principle of inclusion-exclusion: Students who like at least one subject = Like Math + Like Physics - Like both = 18 + 12 - 8 = 22 Students who like neither subject = Total - Like at least one = 30 - 22 = 8 P(likes neither) = Students who like neither / Total students = 8/30 = 4/15 Therefore, the probability is 4/15 or 0.267

Question Type

short_answer

Answer Structure

• Line 1: State given information [0.5 marks]
• Line 2: Apply inclusion-exclusion principle [1 mark]
• Line 3: Calculate students liking neither [1 mark]
• Line 4: Calculate probability [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Double counting students
• Not using inclusion-exclusion principle
• Arithmetic errors

Key Phrases To Include

• inclusion-exclusion
• at least one
• neither subject

Marks

3

Topic

Permutations and Combinations

Difficulty

medium

Template Id

T7

Examiner Tip

List out each letter and its frequency clearly before applying the formula

Model Answer

Given: Word STATISTICS To Find: Number of arrangements Solution: Total letters = 10 Repeated letters: - S appears 3 times - T appears 3 times - A appears 1 time - I appears 2 times - C appears 1 time Number of arrangements = n!/(n₁! × n₂! × n₃! × ...) = 10!/(3! × 3! × 1! × 2! × 1!) = 3,628,800/(6 × 6 × 1 × 2 × 1) = 3,628,800/72 = 50,400 Therefore, there are 50,400 different arrangements.

Question Type

short_answer

Answer Structure

• Line 1: Count total letters and repetitions [1 mark]
• Line 2: Apply permutation formula for repeated objects [1 mark]
• Line 3: Calculate final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Missing repeated letters
• Wrong formula
• Calculation errors

Key Phrases To Include

• repeated letters
• n!/(n₁! × n₂!...)
• arrangements

Marks

2

Topic

Measures of Dispersion

Difficulty

easy

Template Id

T8

Examiner Tip

Always scan the data set completely to ensure you find the true maximum and minimum

Model Answer

Given: Data = {15, 22, 18, 30, 25, 12, 28} To Find: Range Solution: Maximum value = 30 Minimum value = 12 Range = Maximum value - Minimum value Range = 30 - 12 = 18 Therefore, the range is 18.

Question Type

short_answer

Answer Structure

• Line 1: Identify maximum and minimum values [1 mark]
• Line 2: Apply range formula and calculate [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incorrect max/min identification
• Arithmetic error in subtraction

Key Phrases To Include

• maximum value
• minimum value
• Range = Max - Min

Marks

3

Topic

Conditional Probability

Difficulty

medium

Template Id

T9

Examiner Tip

Remember that 'without replacement' changes the total number of balls for the second draw

Model Answer

Given: 5 red balls, 3 blue balls (total = 8) Two balls drawn without replacement To Find: P(both balls are red) Solution: First draw: P(red) = 5/8 After drawing one red ball: 4 red balls remain out of 7 total Second draw: P(red|first red) = 4/7 P(both red) = P(first red) × P(second red|first red) = (5/8) × (4/7) = 20/56 = 5/14 Therefore, the probability that both balls are red is 5/14.

Question Type

short_answer

Answer Structure

• Line 1: State initial conditions [0.5 marks]
• Line 2: Calculate probability for first draw [0.5 marks]
• Line 3: Calculate conditional probability for second draw [1 mark]
• Line 4: Apply multiplication rule [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not accounting for replacement
• Wrong conditional probability
• Not simplifying fraction

Key Phrases To Include

• without replacement
• conditional probability
• multiplication rule

Marks

1

Topic

Permutations and Combinations

Difficulty

easy

Template Id

T10

Examiner Tip

Remember that n! means n × (n-1) × (n-2) × ... × 2 × 1

Model Answer

Number of arrangements = 6! = 720

Question Type

very_short_answer

Answer Structure

• Apply factorial formula and calculate [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong calculation of factorial
• Not using factorial notation

Key Phrases To Include

• 6!
• factorial

Marks

5

Topic

Binomial Distribution

Difficulty

hard

Template Id

T11

Examiner Tip

Always identify the distribution type first, then set up the formula with correct parameters

Model Answer

Given: n = 4 days, p = 0.3 (probability of rain), q = 1 - p = 0.7 To Find: P(exactly 2 rainy days) Solution: This follows binomial distribution with parameters n = 4, p = 0.3 For binomial distribution: P(X = r) = ⁿCᵣ × pʳ × q^(n-r) Where: n = 4, r = 2, p = 0.3, q = 0.7 Step 1: Calculate ⁴C₂ ⁴C₂ = 4!/(2! × 2!) = 24/(2 × 2) = 6 Step 2: Calculate p² p² = (0.3)² = 0.09 Step 3: Calculate q^(n-r) q² = (0.7)² = 0.49 Step 4: Apply the formula P(X = 2) = 6 × 0.09 × 0.49 = 6 × 0.0441 = 0.2646 Therefore, the probability is 0.2646 or 26.46%.

Question Type

long_answer

Answer Structure

• Step 1: Identify binomial distribution parameters [1 mark]
• Step 2: Calculate combination ⁴C₂ [1 mark]
• Step 3: Calculate p² and q² [1 mark]
• Step 4: Apply binomial formula [1 mark]
• Step 5: Final calculation and answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not recognizing binomial distribution
• Errors in combination calculation
• Wrong formula application

Key Phrases To Include

• binomial distribution
• ⁿCᵣ × pʳ × q^(n-r)
• parameters n, p, q

Marks

4

Topic

Statistics of Grouped Data

Difficulty

hard

Template Id

T12

Examiner Tip

Always create the cumulative frequency table systematically and double-check your median class identification

Model Answer

Given: Frequency distribution table To Find: Median Solution: Step 1: Prepare cumulative frequency table Class | Frequency | Cumulative Frequency 0-10 | 5 | 5 10-20 | 8 | 13 20-30 | 12 | 25 30-40 | 10 | 35 40-50 | 5 | 40 Total frequency (N) = 40 Step 2: Find median position Median position = N/2 = 40/2 = 20th observation Step 3: Identify median class The 20th observation lies in class 20-30 (since cf = 25 > 20) Step 4: Apply median formula Median = l + [(N/2 - cf)/f] × h Where: l = 20, N/2 = 20, cf = 13, f = 12, h = 10 Median = 20 + [(20 - 13)/12] × 10 = 20 + (7/12) × 10 = 20 + 5.83 = 25.83 Therefore, the median is 25.83.

Question Type

long_answer

Answer Structure

• Step 1: Create cumulative frequency table [1 mark]
• Step 2: Find N/2 position [1 mark]
• Step 3: Identify median class [1 mark]
• Step 4: Apply median formula [1 mark]

Scoring Breakdown

Common Mark Deductions

• Errors in cumulative frequency
• Wrong median class
• Formula application errors

Key Phrases To Include

• cumulative frequency
• median class
• Median = l + [(N/2 - cf)/f] × h

Marks

2

Topic

Basic Probability

Difficulty

easy

Template Id

T13

Examiner Tip

List all favorable outcomes systematically - consider (a,b) and (b,a) as different outcomes

Model Answer

Given: Two dice thrown simultaneously To Find: P(sum = 8) Solution: Total possible outcomes = 6 × 6 = 36 Favorable outcomes for sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) Number of favorable outcomes = 5 P(sum = 8) = Favorable outcomes/Total outcomes = 5/36 Therefore, the probability is 5/36.

Question Type

short_answer

Answer Structure

• Line 1: Calculate total possible outcomes [0.5 marks]
• Line 2: List favorable outcomes [1 mark]
• Line 3: Apply probability formula [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Missing some favorable outcomes
• Wrong total outcomes
• Not listing outcomes systematically

Key Phrases To Include

• total outcomes = 36
• favorable outcomes
• ordered pairs

Marks

2

Topic

Permutations and Combinations

Difficulty

medium

Template Id

T14

Examiner Tip

Remember: combinations for selection (order doesn't matter), permutations for arrangement (order matters)

Model Answer

Given: Total people = 10, Committee size = 4 To Find: Number of ways to select committee Solution: This is a combination problem (order doesn't matter) Number of ways = ¹⁰C₄ = 10!/(4! × 6!) = (10 × 9 × 8 × 7)/(4 × 3 × 2 × 1) = 5040/24 = 210 Therefore, there are 210 ways to select the committee.

Question Type

short_answer

Answer Structure

• Line 1: Identify as combination problem [0.5 marks]
• Line 2: Apply combination formula [1 mark]
• Line 3: Calculate final answer [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Using permutation instead of combination
• Calculation errors
• Wrong formula

Key Phrases To Include

• combination
• ¹⁰C₄
• order doesn't matter

Marks

4

Topic

Normal Distribution

Difficulty

hard

Template Id

T15

Examiner Tip

Always convert to standard normal first, then use the symmetry property: Φ(-z) = 1 - Φ(z)

Model Answer

Given: μ = 160 cm, σ = 5 cm Heights are normally distributed To Find: P(155 < X < 165) Solution: Step 1: Standardize the values using Z = (X - μ)/σ For X = 155: Z₁ = (155 - 160)/5 = -5/5 = -1 For X = 165: Z₂ = (165 - 160)/5 = 5/5 = 1 Step 2: Find the required probability P(155 < X < 165) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) = Φ(1) - Φ(-1) Step 3: Use standard normal table Φ(1) = 0.8413 Φ(-1) = 0.1587 Step 4: Calculate final probability P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826 Therefore, the probability is 0.6826 or 68.26%.

Question Type

long_answer

Answer Structure

• Step 1: Convert to standard normal using Z-scores [1 mark]
• Step 2: Set up probability expression [1 mark]
• Step 3: Use standard normal table values [1 mark]
• Step 4: Calculate final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incorrect Z-score calculation
• Wrong probability setup
• Table reading errors

Key Phrases To Include

• standardize
• Z-score
• standard normal table
• Φ(z)