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GNDU Question Paper-2024
B.Com 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Max. Marks: 100
Note : Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. (a) Define Statistics. Discuss the functions of Statistics.
(b) From the following frequency distribution, calculate the mean frequency x when the
value of median is 86.
Classes
Frequency
45 – 50
2
50 – 60
1
60 – 70
6
70 – 80
6
80 – 90
x
90 – 100
12
100 – 110
5
2. Calculate Mean, Median and Mode for the following table :
No. of days absent
No. of Students
Less than 5
29
Less than 10
239
Less than 15
469
Less than 20
584
Less than 25
634
Less than 30
644
Less than 35
650
Less than 40
665
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SECTION-B
3. Calculate Quartile Deviation, Mean Deviation, Standard Deviation and Coefficient of
Variation from the following data :
Income (Rs.)
No. of Families
Less than
700
12
800
30
900
50
1000
75
1100
110
1200
120
4. Ten students obtained the following percentage of marks in English in the Internal
Assessment (x) and University Examination (y). Calculate Karl Pearson's coefficient of
correlation.
x
56
60
75
84
47
98
59
44
39
46
y
45
52
90
85
40
85
50
60
32
51
SECTION-C
5. What are Index Numbers ? Differentiate between simple and weighted index numbers.
Explain the importance of weighting in the construction of index numbers. Enumerate the
methods of weighting a price index and discuss their relative merits and demerits.
6. Calculate the Consumer price index number by (i) Aggregate expenditure method and
(ii) Family budget method for the year 2023 taking 2019 as the base.
Commodity
Quantity
Price (Rs.)
2019
2023
Wheat
2 quintals
50
75
Rice
25 kg
100
120
Sugar
10 kg
50
120
Pure Ghee
5 kg
10
10
Milk
5 kg
3
5
Oil
25 kg
200
200
Clothing
25 metres
4
5
Fuel
4 quintals
2
Rent
1 house
8
10
20
25
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SECTION-D
7. From the following data, estimate the trend values for 2024 by (i) least square method
and (ii) 4-yearly moving average.
Year
Sales (in Lakhs)
2013
2014
2015
120
2016
250
2017
240
2018
160
320
8. (a) A bag contains six red balls and six black balls. A ball is picked at random from the
bag and not replaced. A second ball is then picked. Calculate the following probabilities: (i)
The second ball is red, given that the first is red. (ii) Both the balls are red and the first is
red, i.e. the balls are of different colours.
(b) Two events D and E are found to have the following probability relationships : P(D) =
1/2, P(E) = 1/4 and P(D or E) = 1/2. Calculate the following probabilities : (i) P(D and E), (ii)
P(D/E) and (iii) P(E/D)
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GNDU Answer Paper-2024
B.Com 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Max. Marks: 100
Note : Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. (a) Define Statistics. Discuss the functions of Statistics.
(b) From the following frequency distribution, calculate the mean frequency x when the
value of median is 86.
Classes
Frequency
45 – 50
2
50 – 60
1
60 – 70
6
70 – 80
6
80 – 90
x
90 – 100
12
100 – 110
5
Ans: Imagine you are standing on the edge of a bustling city market. You see thousands of
people, countless stalls, and an endless stream of numbers floating everywhere—prices,
quantities, ages, sales, and even temperatures. To make sense of this chaos, to find
patterns, and to make decisions wisely, you need a magical lens that organizes these
numbers and tells their story. That magical lens is Statistics.
Part (a) – Definition and Functions of Statistics
Statistics is like a trusted storyteller who collects information, organizes it neatly, and
interprets it to reveal hidden truths. More formally, Statistics is a branch of mathematics
that deals with the collection, organization, analysis, interpretation, and presentation of
numerical data.
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In simpler words, statistics helps us turn raw numbers into meaningful information so that
we can understand patterns, make predictions, or take decisions. Without statistics,
numbers are like scattered puzzle pieces with no picture to form.
Now, let’s look at what statistics really does—its functions:
1. Collection of Data:
Think of a researcher or a scientist trying to study the eating habits of students in a
university. The first step is to gather data: surveys, questionnaires, observations. This
function of statistics is all about gathering information from the world around us.
No data, no story!
2. Organization of Data:
Once the data is collected, it often looks messy, like a pile of clothes after laundry
day. You need to fold and arrange it, maybe by sorting students by age or meals
consumed per day. In statistics, this is done using tables, charts, and frequency
distributions. Organization helps us see the structure in data, making it easier to
understand.
3. Presentation of Data:
Imagine you want to show your findings to your teacher or colleagues. Raw numbers
in a notebook won’t impress anyone. You need bar charts, pie charts, histograms, or
graphs. Statistics helps present data visually so that the story of the numbers is clear
and easily interpretable.
4. Analysis of Data:
Now comes the exciting part. Once organized and presented, we analyze the data to
find trends, patterns, and relationships. This can involve calculating averages,
percentages, correlations, or other measures that summarize the data meaningfully.
Analysis is where statistics begins to extract wisdom from numbers.
5. Interpretation of Data:
Analysis alone is not enough. You need to interpret the numbers: what do they
mean in real life? If the average marks of students is 75%, does it indicate a strong
performance, or are the exams too easy? Interpretation connects numbers to reality.
6. Prediction or Forecasting:
One of the most powerful functions of statistics is to predict future trends.
Businesses forecast sales, meteorologists predict weather, and doctors predict
disease outbreaks—all using statistical data.
7. Decision-Making:
Finally, statistics helps in making informed decisions. Governments plan policies,
companies decide investments, and individuals make choices based on statistical
reasoning. Without statistics, decisions are mostly guesses.
To sum up, statistics is not just about numbers—it’s a bridge between data and decision-
making, between observation and knowledge. It turns chaos into clarity, confusion into
comprehension.
Part (b) – Solving the Frequency Problem
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Now, let’s step into the shoes of a detective, solving a number puzzle. We are given a
frequency distribution and asked to find the mean frequency x if the median is 86.
Here’s the table again:
Classes
Frequency (f)
45 – 50
2
50 – 60
1
60 – 70
6
70 – 80
6
80 – 90
x
90 – 100
12
100 – 110
5
We are given that the median = 86.
Step 1: Recall the formula for median in grouped data
For a grouped frequency distribution, the median is calculated using the formula:
Median
Where:
• L = lower boundary of median class
• N = total frequency (sum of all frequencies)
• CF = cumulative frequency before median class
• f_m = frequency of median class
• h = class width
Step 2: Determine the median class
The median class is the one that contains the median value 86. Looking at the classes:
• 80 – 90 contains 86 → so median class = 80 – 90
• Lower boundary of median class, L = 80
• Class width, h = 10
Step 3: Express total frequency
Let’s sum the frequencies:
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Step 4: Cumulative frequency before median class
Cumulative frequency before 80 – 90 class:
Frequency of median class: f_m = x
Step 5: Apply the median formula
Median
Plugging in the values:
Simplify step by step:
1.
2. Subtract CF = 15 →
So formula becomes:
Divide both sides by 10:
Multiply both sides by x:
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Convert 0.6x to fraction: 0.6x = 3x/5
Multiply all terms by 10 to remove denominators:
So the frequency of 80 – 90 class, x = 10
Step 6: Calculate mean (optional but usually part of such questions)
For grouped data, mean is calculated using:
Where m_i = midpoints of classes, f_i = frequencies
Step 6a: Find midpoints (m_i):
Classes
Frequency (f)
Midpoint (m)
f × m
45 – 50
2
47.5
95
50 – 60
1
55
55
60 – 70
6
65
390
70 – 80
6
75
450
80 – 90
10
85
850
90 – 100
12
95
1140
100 – 110
5
105
525
,
Mean
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So the mean = 83.45, which fits well with our median 86.
Wrapping It Up
Let’s take a step back. In this story, we saw statistics in action:
1. First, we understood what statistics is—a magical lens that turns messy numbers
into meaningful patterns.
2. We learned its functions—collection, organization, presentation, analysis,
interpretation, forecasting, and decision-making. Each function is like a chapter in
the story of numbers, from raw data to wise decisions.
3. Then, we tackled a real-life frequency problem. Like detectives, we:
o Identified the median class,
o Applied the median formula,
o Solved for the missing frequency, and
o Verified the result by calculating the mean.
This approach shows that statistics is not scary numbers—it’s a tool to understand the
world. It connects theory with practical applications, reasoning with real data, and curiosity
with logical thinking.
Answer Summary:
(a) Definition of Statistics:
Statistics is the branch of mathematics dealing with collection, organization, analysis,
interpretation, and presentation of numerical data.
Functions of Statistics:
• Collection of data
• Organization of data
• Presentation of data
• Analysis of data
• Interpretation of data
• Forecasting and prediction
• Decision-making
(b) Frequency Problem:
• Median class = 80 – 90, L = 80, h = 10
• Cumulative frequency before median = 15
• Median formula gives x = 10
• Mean ≈ 83.45
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Statistics, thus, allows us to transform a simple frequency table into meaningful insights
about central tendencies and distributions—a true story of numbers!
2. Calculate Mean, Median and Mode for the following table :
No. of days absent
No. of Students
Less than 5
29
Less than 10
239
Less than 15
469
Less than 20
584
Less than 25
634
Less than 30
644
Less than 35
650
Less than 40
665
Ans: Late at night, the principal asks you to analyze attendance data. You open the register
and find a neat “less than” table: how many students were absent for less than 5 days, less
than 10, less than 15, and so on. It’s a staircase of numbers, but you need three clear
answers for the report: mean, median, and mode. In other words, what’s the average
absence, the middle point of absence, and the most typical absence? This is a classic
grouped-data story. Let’s turn those cumulative steps into a clear picture and then read the
three central characters of the data: mean, median, and mode.
Given cumulative (“less than”) frequency table
No. of days absent (less than)
No. of students
Less than 5
29
Less than 10
239
Less than 15
469
Less than 20
584
Less than 25
634
Less than 30
644
Less than 35
650
Less than 40
665
Total students .
Converting “less than” cumulative data into class frequencies
To use standard formulas, we first convert cumulative counts into class-wise frequencies.
The natural classes here (width 5) are:
• 0–5, 5–10, 10–15, 15–20, 20–25, 25–30, 30–35, 35–40
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We get each class frequency by successive differences of the cumulative figures.
Class interval
Frequency
Class mark
0–5
5–10
10–15
15–20
20–25
25–30
30–35
35–40
• Check: . Perfect.
Mean of grouped data
For grouped data with class marks
and frequencies
, the mean is:
Compute
:
•
•
•
•
•
•
•
•
Sum:
Total . So:
• Direct answer: The mean number of days absent is about 13.07 days.
Median of grouped data
For the median in grouped data, we identify the median class using . Then use:
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Median
• Labels:
o : lower class boundary of the median class
o : class width
o : frequency of the median class
o
: cumulative frequency before the median class
o
Step-by-step:
• Half of total:
• Cumulative frequencies (up to each class):
o up to 0–5: 29
o up to 5–10: 239
o up to 10–15: 469
• The first cumulative exceeding is , so the median class is 10–15.
• Parameters:
o Class boundaries (continuous) for 10–15 are to , so
o
o Frequency of median class:
o Cumulative before median class:
Plug in:
Median
Median
• Direct answer: The median number of days absent is about 11.53 days.
Mode of grouped data
For the mode (most typical value) in grouped data, identify the modal class (highest
frequency) and use:
Mode
• Labels:
o : lower class boundary of the modal class
o
: frequency of the modal class
o
: frequency of the class preceding the modal class
o
: frequency of the class following the modal class
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o : class width
Step-by-step:
• Frequencies: the largest is (class 10–15), so modal class is 10–15.
• Parameters:
o
o
o
(class 5–10)
o
(class 15–20)
o
Plug in:
Mode
Mode
• Direct answer: The mode number of days absent is about 10.24 days.
Interpretations that make sense to an examiner
• Mean ≈ 13.07 days: On average, students were absent about two and a half weeks in
the period considered. Because mean uses all frequencies and class marks, it is
pulled by the tail of higher absences (e.g., up to 40 days).
• Median ≈ 11.53 days: Half the students were absent less than about 11.5 days and
half more than that. Median is less sensitive to extreme absences; it tells you the
central tendency in terms of position.
• Mode ≈ 10.24 days: The most typical absence (peak of the distribution) sits just over
10 days. This tells where the data “clusters,” reflecting the modal class 10–15.
Together, they sketch the story: most students cluster around approximately 10 days of
absence; half lie below roughly 11.5; the average is slightly higher, around 13, indicating a
right-skew from those with longer absences.
Why we used class boundaries and class marks
• Class marks
let us estimate each group’s central value to compute the mean:
lower limit upper limit
• Continuous class boundaries (e.g., 9.5–14.5 instead of 10–15) make median and
mode formulas accurate for grouped data by removing artificial gaps. With integer
days, values recorded as 10 actually cover 9.5–10.5 on a continuous scale, and so on.
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A clean, stepwise summary for quick revision
• Convert cumulative to class frequency:
o 0–5: 29
o 5–10: 210
o 10–15: 230
o 15–20: 115
o 20–25: 50
o 25–30: 10
o 30–35: 6
o 35–40: 15
• Mean:
o Class marks
: 2.5, 7.5, 12.5, 17.5, 22.5, 27.5, 32.5, 37.5
o
,
o days
• Median:
o , median class: 10–15
o , , ,
o Median days
• Mode:
o Modal class: 10–15
o , ,
,
,
o Mode days
A short narrative to close the loop
Imagine lining up all 665 students by their number of absent days. The student in the very
middle stands at around 11.5 days. Look around that center: you see many clustered a little
over 10 days—that’s your mode. And if you calculate the overall average by weighing every
group, you get a slightly higher number, 13.07, because some students on the far right of
the line stayed absent much longer. The three measures don’t just crunch numbers; they
tell a story—typical absence near 10 days, central position near 11.5, and an average
nudged up to 13 by longer absences. That’s data speaking clearly.
Final answers
• Mean: Approximately 13.07 days
• Median: Approximately 11.53 days
• Mode: Approximately 10.24 days
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SECTION-B
3. Calculate Quartile Deviation, Mean Deviation, Standard Deviation and Coefficient of
Variation from the following data :
Income (Rs.)
No. of Families
Less than
700
12
800
30
900
50
1000
75
1100
110
1200
120
Ans: Imagine we are social scientists exploring a small town. We are curious about how
families earn, how incomes are distributed, and how much variability exists between rich
and poor families. To understand this town’s economic landscape, we have collected a
dataset about families’ monthly incomes. Our mission is to calculate four important
measures: Quartile Deviation, Mean Deviation, Standard Deviation, and Coefficient of
Variation. Each of these measures tells a unique story about the income distribution.
Here is the dataset provided:
Income (Rs.)
No. of Families
Less than 700
12
Less than 800
30
Less than 900
50
Less than 1000
75
Less than 1100
110
Less than 1200
120
At first glance, these numbers might seem like mere figures, but every number tells a story.
For example, “Less than 700, 12 families” shows that 12 families earn less than Rs. 700,
representing the poorer section of the town. As we move up the incomes, more families fall
under higher brackets, revealing the economic spread.
Step 1: Organizing the data
To calculate any statistical measure, we first need to create a frequency distribution table
with class intervals. Right now, the data is in “less than” format. Let’s convert it to actual
income ranges (class intervals) for clarity:
Income Range (Rs.)
Frequency (f)
600–700
12
700–800
18
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800–900
20
900–1000
25
1000–1100
35
1100–1200
10
Here, I assumed the lower class limit as 600 to start the first interval. Now we have a clear
picture: 12 families earn between Rs. 600–700, 18 between 700–800, and so on. This is our
foundation for all calculations.
Step 2: Finding the Quartile Deviation
The Quartile Deviation (QD) is a measure of spread that tells us about the middle 50% of
data. Think of it as finding the “comfortable zone” of incomes in the town, ignoring extreme
highs and lows.
To calculate QD, we need:
1. First Quartile (Q1) – the income below which 25% of families fall.
2. Third Quartile (Q3) – the income below which 75% of families fall.
The formula for QD is:
Step 2a: Locate Q1
• Total families (N) = 120
• Position of Q1 =
th family
Now, we find the class containing the 30th family:
Class
Cumulative Frequency
600–700
12
700–800
12+18 = 30
So, the 30th family falls in 700–800.
Using the quartile formula:
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Where:
• L = lower limit of Q1 class = 700
• CF = cumulative frequency before Q1 class = 12
• f = frequency of Q1 class = 18
• h = class width = 100
Hmm… let's pause. That seems a bit too high; actually, the formula should be:
Position of Q1 – CF
Yes, it’s correct! Q1 = 800 Rs.
Step 2b: Locate Q3
• Position of Q3 =
th family
Cumulative frequency table shows:
Class
CF
900–1000
75
1000–1100
110
So, the 90th family is in 1000–1100.
Step 2c: Quartile Deviation
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So, Quartile Deviation ≈ Rs. 121.43.
This tells us that the “middle 50%” of families have incomes spread roughly 121 Rs. above
and below the median.
Step 3: Calculating Mean Deviation
Mean Deviation (MD) measures the average distance of incomes from the mean. It tells us:
“On average, how far is a family’s income from the town’s average?”
Formula:
Where:
• f = frequency
• x = mid-point of class
• = mean income
• N = total families
Step 3a: Find class mid-points (x):
Class
f
Mid-point (x)
600–700
12
650
700–800
18
750
800–900
20
850
900–1000
25
950
1000–1100
35
1050
1100–1200
10
1150
Step 3b: Find Mean ()
Let’s calculate step by step:
1. 12 × 650 = 7800
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2. 18 × 750 = 13500
3. 20 × 850 = 17000
4. 25 × 950 = 23750
5. 35 × 1050 = 36750
6. 10 × 1150 = 11500
Sum: 7800 + 13500 = 21300
21300 + 17000 = 38300
38300 + 23750 = 62050
62050 + 36750 = 98800
98800 + 11500 = 110300
So, Mean Income ≈ Rs. 919.17
Step 3c: Find Mean Deviation
We calculate for each class:
| Class | f | x | |x - | | f * |x - | |
|-------|---|---|----------------|------------------|
| 600–700 | 12 | 650 | 269.17 | 3229.97 |
| 700–800 | 18 | 750 | 169.17 | 3045.06 |
| 800–900 | 20 | 850 | 69.17 | 1383.40 |
| 900–1000 | 25 | 950 | 30.83 | 770.75 |
| 1000–1100 | 35 | 1050 | 130.83 | 4579.05 |
| 1100–1200 | 10 | 1150 | 230.83 | 2308.30 |
So, Mean Deviation ≈ Rs. 127.64
This tells us that on average, a family’s income deviates about Rs. 128 from the mean,
which is quite reasonable.
Step 4: Calculating Standard Deviation (SD)
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The Standard Deviation gives a more precise picture of variability because it squares the
deviations. Think of it as measuring how “spread out” the incomes are.
Formula:
We calculate
for each class:
Class
f
x
x -
(x - )²
f*(x - )²
600–700
12
650
-269.17
72483.81
869805.7
700–800
18
750
-169.17
28593.44
514681.92
800–900
20
850
-69.17
4784.48
95689.6
900–1000
25
950
30.83
950.65
23766.25
1000–1100
35
1050
130.83
17116.10
598063.5
1100–1200
10
1150
230.83
53382.08
533820.8
Sum = 869805.7 + 514681.92 = 1384487.62
1384487.62 + 95689.6 = 1480177.22
1480177.22 + 23766.25 = 1503943.47
1503943.47 + 598063.5 = 2102006.97
2102006.97 + 533820.8 ≈ 2635827.77
So, Standard Deviation ≈ Rs. 148.21
This confirms the income variability is slightly higher than the mean deviation, which is
expected since SD gives more weight to extreme differences.
Step 5: Calculating Coefficient of Variation (CV)
Finally, the Coefficient of Variation is like putting variability into perspective relative to the
mean. It tells us the degree of income variation in percentage terms:
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This means income varies around 16% relative to the average income—a moderate level of
inequality in this town.
Step 6: The Story in Numbers
Now, let’s wrap up the story these numbers tell:
• Quartile Deviation (121.43 Rs.) shows the spread of the middle 50% of families.
Most families are clustered within about Rs. 120 around the median.
• Mean Deviation (127.64 Rs.) tells us that, on average, families’ incomes deviate
about Rs. 128 from the mean.
• Standard Deviation (148.21 Rs.) captures the spread with more sensitivity to
extremes, showing some families earn far above or below the average.
• Coefficient of Variation (16.12%) gives a sense of relative variability, letting us
compare with other towns or time periods.
Together, these measures give a complete, human-readable picture of income distribution
in the town. It’s like peeking into the town’s economic heart: we see where most families
stand, how unequal the spread is, and how comfortable the middle class is.
Answer (Summary Table):
Measure
Value (Rs.) / %
Quartile Deviation (QD)
121.43
Mean Deviation (MD)
127.64
Standard Deviation (SD)
148.21
Coefficient of Variation (CV)
16.12%
4. Ten students obtained the following percentage of marks in English in the Internal
Assessment (x) and University Examination (y). Calculate Karl Pearson's coefficient of
correlation.
x
56
60
75
84
47
98
59
44
39
46
y
45
52
90
85
40
85
50
60
32
51
Ans: Picture an English teacher in Amritsar, curious about whether her internal assessment
marks really forecast how students will perform in the university exam. Ten students have
two percentages each: x for internal assessment, y for university examination. The teacher
wants one number that tells the strength and direction of their relationship. This number is
Karl Pearson’s coefficient of correlation, denoted by r. If r is near +1, it means strong
positive alignment: higher internal marks tend to accompany higher university marks. If r is
near −1, it’s a strong negative alignment. If r is near 0, there’s no linear pattern. Let’s
compute r step by step and interpret what it says for these students.
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Data given
• x: 56, 60, 75, 84, 47, 98, 59, 44, 39, 46
• y: 45, 52, 90, 85, 40, 85, 50, 60, 32, 51
We will use the direct (raw sum) formula for Pearson’s r:
where is the number of paired observations.
Organizing the calculation
To keep the arithmetic clean, compute the following totals:
• Σx, Σy
• Σx², Σy²
• Σxy
A working table helps ensure accuracy:
Student
x
y
x²
y²
xy
1
56
45
3136
2025
2520
2
60
52
3600
2704
3120
3
75
90
5625
8100
6750
4
84
85
7056
7225
7140
5
47
40
2209
1600
1880
6
98
85
9604
7225
8330
7
59
50
3481
2500
2950
8
44
60
1936
3600
2640
9
39
32
1521
1024
1248
10
46
51
2116
2601
2346
Now sum each column:
• Σx: 608
• Σy: 590
• Σx²: 40284
• Σy²: 38604
• Σxy: 38924
• n: 10
Applying the formula
• Numerator:
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• Denominator parts:
• Pearson’s r:
Interpretation
• Strength: indicates a strong positive linear relationship.
• Direction: Positive means students who scored higher in internal assessment tended
also to score higher in the university exam.
• Practical sense: Internal assessment scores are a fairly reliable indicator of university
performance for this group, though not perfect.
A clear, student-friendly narrative of what r means
• Imagine a scatterplot: Each student is a point with x (internal) on the horizontal axis
and y (university) on the vertical axis. If you draw a best-fit straight line through
these points, a strong positive r means the points lie close to that line and slope
upward: as x increases, y tends to increase.
• Why not 1.00? Real data includes variation—study habits change, exam difficulty
differs, stress or health affects performance. So while the trend is strong, it’s not
perfect.
• Why not 0.00? Because the two exams test overlapping skills in English—
comprehension, expression, grammar—so higher internal marks reasonably align
with higher university marks.
Common pitfalls avoided
• Using the wrong formula: We used the direct raw-sum formula suitable for paired
data without grouping.
• Mixing up sums: It’s crucial to compute Σx, Σy, Σx², Σy², Σxy carefully. A single
arithmetic slip can change r noticeably.
• Forgetting n: The raw-sum formula heavily depends on the correct value of .
Alternative computation routes (and why this one is best here)
• Deviation-from-mean method: Compute deviations
and
, then use the
formula
. It yields the same result but requires additional mean
calculations.
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• Assumed-mean method: Useful when numbers are large; you pick convenient
assumed means to reduce arithmetic load. Not necessary here since the dataset is
small.
• Covariance form:
Cov
. Conceptually elegant, but computationally similar to
the deviation method.
For ten pairs, the direct raw-sum route is fast, transparent, and exam-friendly.
Short, examiner-friendly summary of steps
1. List paired data and compute totals: Σx, Σy, Σx², Σy², Σxy; note .
2. Compute numerator: .
3. Compute denominator parts: , , so
.
4. Calculate r:
.
5. Interpret: Strong positive linear correlation between internal and university English
marks.
Why correlation is useful beyond this problem
• Predictive insight: With a strong positive r, internal assessments can help identify
students likely to do well later, enabling targeted support where needed.
• Quality checks: If internal and university marks had been weakly correlated (say
), it would raise questions about alignment of internal tests with the syllabus or
evaluation standards.
• Decision-making: Administrators can use such analyses to refine teaching strategies,
moderate internal marking, and calibrate difficulty levels for better alignment.
SECTION-C
5. What are Index Numbers ? Differentiate between simple and weighted index numbers.
Explain the importance of weighting in the construction of index numbers. Enumerate the
methods of weighting a price index and discuss their relative merits and demerits.
Ans: Imagine you are the manager of a bustling marketplace. Every day, countless items—
vegetables, fruits, grains, clothes, and gadgets—change in price. Some days, the price of
tomatoes shoots up; other days, the price of rice drops. Now, as a marketplace manager,
you want to know one big thing: how the overall prices of things are changing over time.
Are prices going up steadily, falling, or fluctuating wildly? This is where Index Numbers
come into play.
What are Index Numbers?
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Think of an index number as a summary or a snapshot of change. Instead of looking at
hundreds of individual prices, an index number compresses all that information into one
single figure that tells you whether prices or quantities have increased or decreased over
time.
In technical terms, an index number is a statistical measure designed to show changes in a
variable or a group of related variables over time, or between different locations, relative
to a base period. For example, the Consumer Price Index (CPI) tells us how the prices of a
basket of goods change over time compared to a base year. Similarly, the Wholesale Price
Index (WPI) monitors the prices at the wholesale level.
Think of it like a thermometer for the economy: instead of checking the temperature of
each individual room (prices of individual items), you get an overall reading of whether the
economy is heating up (inflation) or cooling down (deflation).
Simple Index Numbers vs. Weighted Index Numbers
Now, imagine you are making an index for your market. You pick a few items: rice, wheat,
and sugar. There are two ways to create an index: simple and weighted.
1. Simple Index Numbers
A simple index number treats all items equally, no matter how important they are in
people’s daily lives. It’s like saying:
"A rise in the price of sugar is just as important as a rise in the price of rice or wheat."
The formula is straightforward:
Simple Price Index
Price in current period
Price in base period
Example:
• Base year prices: Rice = 50, Wheat = 40, Sugar = 20
• Current year prices: Rice = 60, Wheat = 50, Sugar = 30
Then, the simple price index for each item would be:
• Rice:
• Wheat:
• Sugar:
If we take the average of these, the simple index number is:
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So, overall, prices have risen about 31.67% compared to the base year.
Merit:
• Simple to calculate and understand.
Demerit:
• Not all items are equally important in real life. People spend more on rice than sugar.
Treating them equally can give a misleading picture.
2. Weighted Index Numbers
This is where weighting comes in. Weighted index numbers recognize that not all items are
equally significant. Some items, like rice or electricity, may take a larger portion of a
household budget, while items like spices or stationery may take less.
In a weighted index, each item is assigned a weight proportional to its importance
(consumption, sales, or production). The formula is:
Weighted Index
Price Relative Weight
Weights
Example continuing from above:
• Weights based on household expenditure: Rice = 50, Wheat = 30, Sugar = 20
• Price relatives: Rice = 120, Wheat = 125, Sugar = 150
Weighted Index:
Notice the difference? With weights, the overall index is 127.5, lower than the simple index
of 131.67. Why? Because rice, which has the largest weight, increased less than sugar, so
the weighted average gives a more realistic picture of how price changes affect the
economy.
Importance of Weighting in Index Numbers:
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1. Reflects true economic impact of price changes.
2. Accounts for differences in consumption or production patterns.
3. Helps policymakers and businesses make informed decisions about inflation,
subsidies, or price controls.
Methods of Weighting a Price Index
There are several ways to assign weights, each with its advantages and limitations. Let’s
explore them like characters in our marketplace story.
1. Simple Aggregative Method (Quantity or Value Weights)
• Here, the weight is proportional to the quantity consumed or sold.
• Formula:
Index
Where
= current price,
= base price,
= quantity in base period
Merit:
• Reflects actual consumption patterns.
• Easy to calculate if quantity data is available.
Demerit:
• If quantities change significantly over time, index may be biased.
• Ignores changes in preferences or new goods.
2. Laspeyres Index
• Uses base period quantities as weights.
• Emphasizes past consumption patterns.
Merit:
• Easy to compute using base-year data.
• Good for historical comparisons.
Demerit:
• May overstate inflation if people substitute expensive goods with cheaper ones.
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3. Paasche Index
• Uses current period quantities as weights.
• Reflects present consumption patterns.
Merit:
• Reflects current economic behavior better than Laspeyres.
Demerit:
• Requires more current data.
• May understate inflation if cheaper goods dominate current consumption.
4. Fisher’s Ideal Index
• Geometric mean of Laspeyres and Paasche indices.
• Tries to balance the overstatement of Laspeyres and understatement of Paasche.
Merit:
• Considered most accurate and reliable.
Demerit:
• Computationally complex.
5. Aggregate Expenditure Method
• Weights based on total monetary expenditure on each item.
• More direct measure of impact on consumer budgets.
Merit:
• Reflects real-world financial impact.
Demerit:
• Requires detailed expenditure data, which may not always be available.
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Bringing the Story Together
Imagine you are now reporting to the city mayor about the cost of living. If you used simple
index numbers, the mayor might think inflation is higher than it really is. But using weighted
index numbers, with weights reflecting actual household spending, the report becomes
accurate and actionable.
The choice of method—Laspeyres, Paasche, or Fisher—depends on data availability and the
purpose. For policymakers, a weighted approach is like a compass, guiding them to control
inflation, adjust subsidies, or plan public policies. For businesses, it helps in pricing
strategies or forecasting demand.
Conclusion
To sum it up in our marketplace story:
1. Index Numbers are the “thermometer” of prices and quantities.
2. Simple indices are easy to calculate but treat all items equally, which may mislead.
3. Weighted indices assign importance to each item, reflecting the real economic
impact.
4. Weighting is essential to make indices realistic, relevant, and useful.
5. Different weighting methods—Laspeyres, Paasche, Fisher, or aggregate
expenditure—each have strengths and weaknesses. Choosing the right method
ensures our index is trustworthy and practical.
In short, without weighted index numbers, you’re like a doctor diagnosing a patient by only
looking at one symptom. With weighted indices, you have the full picture—the patient’s
entire health chart. And that’s why economists and statisticians treasure weighted index
numbers.
6. Calculate the Consumer price index number by (i) Aggregate expenditure method and
(ii) Family budget method for the year 2023 taking 2019 as the base.
Commodity
Quantity
Price (Rs.)
2019
2023
Wheat
2 quintals
50
75
Rice
25 kg
100
120
Sugar
10 kg
50
120
Pure Ghee
5 kg
10
10
Milk
5 kg
3
5
Oil
25 kg
200
200
Clothing
25 metres
4
5
Fuel
4 quintals
2
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Rent
1 house
8
10
20
25
Ans: Picture a family in Amritsar planning their monthly budget. They buy wheat and rice,
sugar and milk, pay rent, fill the stove with fuel, and set aside a little for clothing and oil.
Prices change with the seasons and the years, and the family wants to know one simple
thing: by how much has the cost of living gone up since 2019? That single answer is the
Consumer Price Index (CPI). It’s not just a number; it’s a story of how the same basket of
goods costs more or less across time.
You’re given the basket, the quantities the family consumes, and the prices in 2019 and
2023. Your task: compute CPI by two classic methods—Aggregate Expenditure (weighted by
base quantities) and Family Budget (weighted average of price relatives). We’ll turn this
dataset into two clean indices and interpret them. Along the way, we’ll handle a couple of
gaps in the data and show you exactly what needs confirming to finish the calculation.
The dataset and how to read it
• Base year: 2019
• Current year: 2023
• Quantities are base-year quantities, used as weights in both methods.
The table as presented:
• Wheat, 2 quintals; price 2019 = 50; price 2023 = 75
• Rice, 25 kg; 2019 = 100; 2023 = 120
• Sugar, 10 kg; 2019 = 50; 2023 = 120
• Pure Ghee, 5 kg; 2019 = 10; 2023 = 10
• Milk, 5 kg; 2019 = 3; 2023 = 5
• Oil, 25 kg; 2019 = 200; 2023 = 200
• Clothing, 25 metres; 2019 = 4; 2023 = 5
• Fuel, 4 quintals; 2019 = 2; 2023 = [missing]
• Rent, 1 house; 2019 = 8; 2023 = 10
• Last line: “20 25” (appears to be a price pair without a commodity/quantity label)
Two issues we must clarify to finish the numeric CPI:
• The 2023 price for Fuel is missing.
• The final “20 25” entry lacks commodity and quantity. If it is a valid item, we need
both the base-year quantity and the item name; otherwise we should exclude it.
We can still compute partial sums and lay out the complete procedure so the final step
becomes plug-and-play once you provide those missing values.
What CPI is measuring
The Consumer Price Index compares the cost of a fixed basket in the current year to the cost
of the same basket in the base year.
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• Aggregate Expenditure (Laspeyres-type): keeps quantities fixed at base-year levels,
compares total expenditure at current prices to total expenditure at base prices.
• Family Budget (weighted average of price relatives): computes price relatives for
each item and averages them using base-year expenditure as the weight.
Both tell the same story from two angles—one via total rupees spent, the other via relative
price changes weighted by importance.
Method (i): Aggregate expenditure method
Formula
CPI
AE
•
: price in 2019
•
: price in 2023
•
: quantity in 2019 (the fixed basket)
We compute two totals:
• Base-year expenditure
• Current-year expenditure
Step-by-step computation with available items
Let’s multiply each item’s price by its base quantity.
• Wheat:
quintals
o Base:
o Current:
• Rice: kg
o Base:
o Current:
• Sugar: kg
o Base:
o Current:
• Pure Ghee: kg
o Base:
o Current:
• Milk: kg
o Base:
o Current:
• Oil: kg
o Base:
o Current:
• Clothing: metres
o Base:
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o Current:
• Fuel: quintals
o Base:
o Current:
fuel
(2023 price missing)
• Rent: house
o Base:
o Current:
• Unknown last line “20 25”: cannot include without quantity and item name.
Known partial totals
• Base total (excluding unknown last line):
• Current total (excluding fuel current price and unknown last line):
fuel
fuel
CPI expression with the current fuel price placeholder
CPI
AE
fuel
Once you provide the 2023 price of fuel (per quintal), plug it into
fuel
. If the last “20 25” line
is a valid item, we’ll add
to 8281 and
to the numerator accordingly; the formula
remains the same.
Method (ii): Family budget method (weighted average of price relatives)
Formula
CPI
FB
where:
•
(price relative for each item)
•
(the base-year expenditure, serving as weight)
This method treats each item’s percentage price change and averages them according to
how much of the budget each item occupies in the base year.
Compute weights W and relatives R
We already have
from the prior step:
• Wheat , Rice , Sugar , Ghee , Milk , Oil , Clothing ,
Fuel , Rent .
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Now compute
:
• Wheat:
• Rice:
• Sugar:
• Pure Ghee:
• Milk:
• Oil:
• Clothing:
• Fuel:
fuel
fuel
(since base price is 2; per quintal)
• Rent:
Multiply R by W and sum
Compute for each known item:
• Wheat:
• Rice:
• Sugar:
• Pure Ghee:
• Milk:
• Oil:
• Clothing:
• Fuel:
fuel
fuel
• Rent:
Sum of weights (excluding unknown last line) is:
Sum of (excluding fuel placeholder and unknown last line):
Add fuel’s term:
fuel
.
CPI expression (Family Budget)
CPI
FB
fuel
Note that this formula currently yields a “scaled index” because we used as percentage
and in rupees. To convert to the usual 100-base index, observe that the denominator is
and the numerator is ; that’s already the standard formula, so the result is the
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CPI level (e.g., 128 means 28% higher than base). Numerically, once
fuel
is provided, you
divide and read the index.
If the “20 25” line is a valid item with base quantity
?
, you’ll add:
• To :
?
?
• To :
?
?
?
Interpreting the two indices
• Aggregate Expenditure CPI tells you directly: “If we bought the same quantities as in
2019, how many rupees would we need at 2023 prices compared to 2019?” It is a
Laspeyres-style index and tends to overstate inflation slightly when consumers
substitute away from items that get relatively more expensive.
• Family Budget CPI refines this idea by taking item-wise percentage changes and
averaging them using base budget shares. It gives similar results and is common in
textbook problems, especially for fixed baskets.
They should be close, and any difference usually arises from rounding and the structure of
the relative changes.
Why some items matter more in the final index
Look at the weights
. Oil has a very large weight (₹5000), rent and rice are
meaningful, while milk and ghee are relatively small. In both methods, items with larger
base expenditure dominate the index. A big price change in a tiny-weight item (like milk
here) moves the CPI less than a small change in a huge-weight item (like oil). That’s exactly
how a real family’s budget behaves.
What we need from you to finalize the numbers
To deliver exact CPI values:
1. The 2023 price of Fuel (per quintal). If, for instance,
fuel
(a common exam-style
change), then:
o Aggregate Expenditure numerator becomes
.
o Family Budget numerator becomes .
2. Clarify the last line “20 25”:
o Is it a commodity (e.g., “Entertainment” or “Miscellaneous”) with base-year
quantity missing? If yes, please share the quantity.
o If it’s a formatting artifact, we will exclude it.
Once you confirm these, I’ll plug the values into the formulas and give you clean CPI
numbers with two decimal places.
If we assume Fuel (2023) price = 3 and “20 25” line is not an item
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Just to illustrate the finish:
• Aggregate Expenditure:
CPI
AE
That means the basket costs about 15.6% more in 2023 than in 2019.
• Family Budget:
CPI
FB
This would imply about 27.6% inflation. However, notice this gap is unusually large
compared with the Aggregate Expenditure result. The reason is that sugar’s price relative
(240) and milk’s (≈166.7) heavily amplify in the weighted relatives method, while the direct
expenditure comparison dampens the effect because oil (with a huge weight) has no price
change. In exam datasets, these two methods usually give closer values when weights are
balanced; here, the oil weight being very large at constant price drives a lower AE CPI.
This contrast is a teachable point: the Family Budget method averages percentage changes,
potentially giving more influence to extreme relatives in small items, while the AE method
tracks rupee spending with large-weight items anchoring the result.
To avoid confusion, I recommend finalizing with your actual Fuel price and clarifying the last
line, then we’ll present both indices side by side and discuss the difference succinctly.
The story behind the numbers
• Wheat and rice have risen—families feel this in everyday meals.
• Sugar has jumped sharply—its high relative inflates the family-budget index notably.
• Milk costs more—a daily staple adding steady pressure.
• Oil and ghee are flat here—this stabilizes the aggregate expenditure index because
oil has a major budget share.
• Rent is up—household essentials seldom go down, and rent’s rise matters even with
a smaller weight.
These movements compose the melody of the CPI: some instruments are louder (big
weights), some play sharper notes (big relatives), and the two methods mix them differently
to produce the final tune.
SECTION-D
7. From the following data, estimate the trend values for 2024 by (i) least square method
and (ii) 4-yearly moving average.
Year
Sales (in Lakhs)
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2013
2014
2015
120
2016
250
2017
240
2018
160
320
Ans: Imagine this: A Tale of Sales Through the Years
Imagine you are the manager of a company called "Lakhs & Co.", which sells innovative
gadgets. You’ve been keeping track of annual sales since 2013. As a careful planner, you
want to predict future sales so that your company doesn’t run out of stock or overproduce.
The table you have is like a diary of your sales:
Year
Sales (in Lakhs)
2013
—
2014
—
2015
120
2016
250
2017
240
2018
160
2019
320
Your task is to estimate the trend value for 2024. Trend analysis is like seeing the “general
mood” of your sales over time, ignoring small ups and downs. There are many ways to do
this, but we focus on two famous methods:
1. Least Squares Method – Think of it as fitting a straight line through your sales diary,
showing the general direction your sales are moving.
2. 4-Yearly Moving Average – Imagine smoothing out the ups and downs by averaging
every 4 years, giving you a calm, flowing line of sales trends.
Let’s explore each method carefully.
Part 1: Trend Estimation Using the Least Squares Method
Step 1: Understanding the Method
The Least Squares Method is like having a rubber band stretched through all your sales
points on a graph. The rubber band tries to come as close as possible to all points,
minimizing the distance between the points and the line. Mathematically, this gives a trend
equation:
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Where:
• = Trend value (sales we want to estimate)
• = Time variable (year coded as 0, ±1, ±2, …)
• = Intercept (starting value)
• = Slope (rate of increase/decrease per year)
We first assign numbers to years to simplify calculation:
Year
Sales (Y)
X (Year Code)
2015
120
-2
2016
250
-1
2017
240
0
2018
160
1
2019
320
2
Why code years? Coding makes the math simple because it centers the years around zero.
Step 2: Calculate and
The formulas are:
Where = number of observations.
1. Compute and
:
Year
X
Y
X·Y
X²
2015
-2
120
-240
4
2016
-1
250
-250
1
2017
0
240
0
0
2018
1
160
160
1
2019
2
320
640
4
•
•
•
Now:
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So the trend equation is:
Step 3: Predict Sales for 2024
To predict 2024, first calculate the X value for 2024. Our central year (X=0) is 2017. Each
step is +1 for each year forward:
• 2017 → X = 0
• 2018 → X = 1
• 2019 → X = 2
• 2020 → X = 3
• 2021 → X = 4
• 2022 → X = 5
• 2023 → X = 6
• 2024 → X = 7
Now plug into :
Estimated trend value for 2024 using Least Squares Method = 435 Lakhs
Part 2: Trend Estimation Using 4-Yearly Moving Average
Step 1: Understanding the Method
The 4-yearly moving average is like smoothing out the rollercoaster of sales. Imagine a
gentle line passing through the yearly peaks and valleys, taking the average of 4 consecutive
years at a time.
The formula is simple:
4-yearly moving average
Sum of sales of 4 consecutive years
It reduces random fluctuations and focuses on the long-term trend.
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Step 2: Calculate 4-Year Moving Averages
We only have 2015–2019, so the first 4-year period is 2015–2018:
MA (2016)
Next 4-year period is 2016–2019:
MA (2017)
Notice the moving averages are centered between the years (like 2016.5 and 2017.5). Since
we want 2024, we can extrapolate the trend from the moving averages.
We calculate the average increase per year from the two moving averages:
• Increase = 242.5 − 192.5 = 50
• Time difference = 1 year (approx)
So trend increases by 50 per year.
From last moving average (2017.5 → 242.5), calculate 2024 (X = 2024 − 2017.5 = 6.5 years
forward):
Estimated trend value for 2024 using 4-Yearly Moving Average = 568 Lakhs (approx)
Notice how this method gives a slightly higher trend because moving averages smooth out
dips and show an upward bias if last years are strong.
Comparing the Two Methods
Method
Estimated Trend 2024
Least Squares Method
435 Lakhs
4-Yearly Moving Average
568 Lakhs
• Least Squares is based on fitting a straight line through all points. It balances all
highs and lows.
• Moving Average focuses on recent periods, smoothing fluctuations, so it is sensitive
to the latest data.
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Think of it as:
• Least Squares → Wise old sage, looks at the whole history and predicts steadily.
• Moving Average → Trendy friend, reacts to latest events and gives a modern
forecast.
Final Thoughts: The Moral of the Story
Predicting trends is not just math; it’s a strategic skill. It helps businesses:
• Avoid underproduction or overproduction
• Forecast budgets
• Plan for long-term growth
Both methods have pros and cons:
• Least Squares: Accurate for long-term trends, less influenced by short-term
fluctuations.
• Moving Average: Smooths out randomness, useful when recent years dominate the
trend.
In real life, analysts often use both methods together to get a balanced view. It’s like having
two advisors: one looks at the whole forest, the other focuses on the path ahead.
Conclusion
So, dear reader, we have successfully traveled through the world of trend analysis:
1. Using Least Squares, the estimated sales trend for 2024 = 435 Lakhs
2. Using 4-Yearly Moving Average, the estimated trend = 568 Lakhs
By understanding the story behind the numbers, this task stops being a boring calculation
and becomes an exciting prediction game. Just like a story of your company’s growth,
trends reveal the secret rhythm of business.
8. (a) A bag contains six red balls and six black balls. A ball is picked at random from the
bag and not replaced. A second ball is then picked. Calculate the following probabilities: (i)
The second ball is red, given that the first is red. (ii) Both the balls are red and the first is
red, i.e. the balls are of different colours.
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(b) Two events D and E are found to have the following probability relationships : P(D) =
1/2, P(E) = 1/4 and P(D or E) = 1/2. Calculate the following probabilities : (i) P(D and E), (ii)
P(D/E) and (iii) P(E/D)
Ans: Imagine you’re supervising a friendly game in class. There’s a bag with red and black
balls, and everyone is pulling two balls one after another. As the game goes on, you start
wondering: what’s the chance the second ball is red if the first one was red? What’s the
chance the two balls are of different colours? Later, you find a separate puzzle about two
events, D and E, with some given probabilities, and you want to know how they overlap and
how one affects the other. These are the kinds of questions that turn probability from dry
numbers into a living story: you, the bag, the draws, and the logic that binds them.
Let’s work through both parts, cleanly and calmly, and build intuition as we go.
Part (a) Two draws from a bag of red and black balls
We have a bag with 6 red balls and 6 black balls. Two balls are picked sequentially without
replacement.
Setup and simple facts
• Total balls initially: 12 (6 red + 6 black).
• Without replacement: The first draw changes the composition of the bag before the
second draw.
We’ll address each required probability, carefully interpreting the phrases and clarifying any
ambiguity.
The second ball is red, given that the first is red
This is a conditional probability question. You already know the first ball is red; what’s the
chance the second is red?
• After first red: Remaining red balls are 5, remaining total is 11.
• Conditional probability:
second red first red
Direct answer: 5/11.
Both balls are red, and the first is red, i.e., the balls are of different colours
This line appears contradictory at first glance: “both the balls are red” versus “the balls are
of different colours” cannot be true simultaneously. In many exam sets, the intended sub-
parts are:
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• Probability both balls are red.
• Probability the balls are of different colours.
We will compute both, and also give the conditional version “second black given first red” to
fully cover the intent.
Both balls are red
Two successive reds from 6 red, 6 black:
• First red:
• Second red after first red:
Therefore,
both red
Direct answer: 5/22.
Balls are of different colours
There are two ways to get different colours:
• Red then Black.
• Black then Red.
Compute each and add.
• Red then Black:
• Black then Red:
Sum:
different colours
Direct answer: 6/11.
Bonus: The second ball is black, given the first is red
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This is the conditional version aligned with “different colours” when the first is known to be
red:
• After first red, the bag has 6 black out of 11 total.
second black first red
This matches the idea of “different colours” under the condition that first was red.
Intuition for Part (a)
• Given first red: You “waste” one red from the bag; fewer reds remain than blacks (5
vs. 6), so the chance the second is red is slightly less than the chance it’s black, which
is why .
• Different colours higher than both red: Because starting balanced (6 reds, 6 blacks),
it’s more likely to get a mixed pair than a double of one colour in two draws.
Part (b) Two events D and E with given relationships
We’re told:
•
•
• or
Recall the identity:
We can use this to find the intersection, then the conditional probabilities and
.
(i) Compute
From the union formula:
Rearrange:
Direct answer: .
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(ii) Compute
Definition:
Plug values:
Direct answer: 1. Interpreted: If E happens, D is certain. So E is a subset of D.
(iii) Compute
Definition:
Direct answer: . Interpreted: If D happens, E happens half the time.
Intuition for Part (b)
• The fact that tells you E can only occur inside D; think of E as living
entirely within D’s region in a Venn diagram.
• Meanwhile, means that within D’s territory, E occupies half the area.
So D is “broader,” and E is nested inside it.
Clean summary answers
• Part (a):
o second red first red
o both red
o different colours
o Bonus: second black first red
• Part (b):
o
o
o
Why this matters beyond the exam
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• Conditional probability captures how knowledge changes chances. In the bag
problem, the first draw reshapes the second draw’s probabilities; in the events
problem, knowing E fully determines D.
• Intersection and union logic keeps multi-event problems honest. Always anchor
your computation to the identity . It prevents
double-counting.
• Interpretation turns numbers into insight. Saying “” is more than
arithmetic—it tells you E is fully inside D. Seeing
and
for the second draw
clarifies why “different colours” are more likely than “both red” in a balanced bag.
“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
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