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GNDU Question Paper-2023
Bachelor of Business Administration
BBA 3
rd
Semester
STATISTICS FOR BUSINESS
Time Allowed: Three Hours Max. Marks: 50
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.(i) If A =
find A
2
-5A + 61.
(ii). If
=
find x.
2. (i) Show that:
= (a-b)(b-c)(c-a).
(ii) Solve the equations:
x - y + z = 4 2x + y - 3z = 0 x + y + z = 2 by Cramer's rule.
SECTION-B
3.(i) Mean of 100 observations is found to be 40. If at the time of computation, two items
were wrongly taken as 30 and 27 instead of 3 and 72, find correct mean.
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(ii) Find Median of the following data:
Mid Points
5
15
25
35
45
Frequency
3
9
8
5
3
(i) What do you mean by Standard Deviation? Write its merits and demerits. 5
(ii) From the following observations, calculate quartile deviation and coefficient of quartile
deviation:
59, 62, 65, 64, 63, 61, 60, 56, 58, 66. 5
SECTION-C
5. Calculate coefficient of Correlation between X and Y for the following data:
X
1
3
4
5
7
8
10
Y
2
6
8
10
14
16
20
6. Discuss the importance of time series analysis. Also discuss its components.
SECTION-D
5 (i) Three coins are tossed simultaneously. Write its sample space. Also find the
probability of getting at least one head.
(ii) The probability of solving specific problem by A and B is 1/2 and 1/3 respectively. If
both A and B try to solve the problem independently, find the probability that the
problem is solved.
8. (i) What is Binomial Probability distribution? Write its important properties.
(ii) Discuss Normal Probability distribution and its main characteristics.
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GNDU Answer Paper-2023
Bachelor of Business Administration
BBA 3
rd
Semester
STATISTICS FOR BUSINESS
Time Allowed: Three Hours Max. Marks: 50
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.(i) If A =
find A
2
-5A + 61.
(ii). If
=
find x.
Ans: (i) A friendly tale of two problems (and how to tame them)
Let’s begin with a picture. Imagine you’re standing in a small workshop where numbers are
the raw materials, and matrices are the machines. You feed a matrix into the machine twice,
mix in some multiples of itself, and combine everything with the identity “safety net.” That’s
exactly what the first problem asks you to do: compute A2−5A+6. Then, in the second
problem, you’re given two square “frames” (determinants) that happen to be equal, and
you’re asked to find the value of xxx that makes that equality true. We’ll handle both with
calm, precise steps—no stress—while keeping the journey enjoyable and crystal clear.
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(ii). (ii). If
=
find x.
Ans:
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2. (i) Show that:
= (a-b)(b-c)(c-a).
(ii) Solve the equations:
x - y + z = 4 2x + y - 3z = 0 x + y + z = 2 by Cramer's rule.
Ans: (i). Imagine three classmates—Ami, Bela, and Chiru—lining up for a photo. The
photographer says, “You three look similar but stand at different places. I want to capture
the difference between where you stand.” That little sentence is the secret of today’s
problems: we’re going to “capture differences.” In algebra, one of the neatest cameras for
differences is the determinant. For (i), we’ll show a famous 3×3 determinant—built from
numbers a,b, c—collapses beautifully into a product of pairwise differences. Then, for (ii),
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we’ll use the same determinant idea (Cramer’s Rule) to solve a system of linear equations in
a crisp, exam-pleasing way.
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(ii). Solve the equations:
x - y + z = 4 2x + y - 3z = 0 x + y + z = 2 by Cramer's rule.
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SECTION-B
3.(i) Mean of 100 observations is found to be 40. If at the time of computation, two items
were wrongly taken as 30 and 27 instead of 3 and 72, find correct mean.
(ii) Find Median of the following data:
Mid Points
5
15
25
35
45
Frequency
3
9
8
5
3
Ans: The Magical Journey of Mean and Median
Every student has faced that moment when numbers start looking like a secret code. They
sit in neat rows, whispering among themselves, and the teacher tells us, “Find the mean!” or
“Find the median!” At first, it feels like we’ve been thrown into a mystery novel where
numbers are the characters and we are the detectives. But if we think of it as a story rather
than a stiff formula, the solution becomes much easier—and actually fun!
Today, we’ll solve two such detective cases:
1. A mystery of the “wrong entries” while calculating the mean of 100 observations.
2. A curious search for the median in a grouped frequency distribution.
But before we dive into calculations, let’s warm up with a small story that will help us
connect to these concepts.
Story Time: The Sweet Shop Mystery
Imagine a sweet shop owner who records how many sweets he sells every day. After a
month, he decides to calculate the average (mean) sales. But while noting down two days’
sales, he accidentally wrote the wrong numbers! Instead of writing “3 boxes” for one day
and “72 boxes” for another, he mistakenly wrote “30 boxes” and “27 boxes.”
When he later calculated the average sales, the result came out wrong. Only after some
checking did he realize the mistake. So, he asked his smart nephew (that’s us, the student!)
to help him find the correct average.
This little situation is exactly the same as our first question. The sweet shop owner is like the
person handling 100 observations, and the wrong entries are the “misrecorded items.”
Now let’s carefully solve this step by step.
Part (i) Correcting the Mean
We are told:
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• Number of observations (n) = 100
• Wrong mean = 40
• Two wrong values taken = 30 and 27
• Correct values should be = 3 and 72
Step 1: Find the total sum according to the wrong calculation.
Mean is basically:
So,
So, the person had wrongly considered the total of all numbers as 4000.
Step 2: Correct the mistake in the sum.
In the wrong sum, 30 and 27 were used. But the correct numbers should have been 3 and
72.
So, we adjust the total like this:
So, the corrected total of all observations = 4018.
Step 3: Calculate the correct mean.
So, the correct mean is 40.18.
Detective case one solved! The average wasn’t 40, but actually 40.18.
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Part (ii) Finding the Median
Now comes the second case. Here, we are asked to find the median of the following data:
| Midpoints (x) | 5 | 15 | 25 | 35 | 45 |
|---------------|---|----|----|----|
| Frequency (f) | 3 | 9 | 8 | 5 | 3 |
Let’s slow down and first understand what median really means.
The Concept of Median (Simple Explanation)
If the mean is like dividing sweets equally among children, then the median is like finding
the child who sits in the exact middle when all children are lined up in order of height.
For example: Suppose 11 children are standing in a row by height. The median height will be
that of the 6th child—because there are 5 shorter children on the left and 5 taller children
on the right.
In grouped data, it’s not so direct because the data is in intervals or midpoints with
frequencies. So, we use a formula to find the exact middle value.
Step 1: Arrange the data in a frequency distribution table
| Midpoints (x) | 5 | 15 | 25 | 35 | 45 |
|---------------|---|----|----|----|
| Frequency (f) | 3 | 9 | 8 | 5 | 3 |
Now, calculate cumulative frequencies (CF):
Midpoints (x)
Frequency (f)
Cumulative Frequency (CF)
5
3
3
15
9
12
25
8
20
35
5
25
45
3
28
So, total observations N=28.
Step 2: Find the median class
Median position is given by:
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So, the 14th observation is the median term.
Looking at the CF column:
• Up to 12 → data covers first 12 observations.
• Up to 20 → data covers observations from 13th to 20th.
Therefore, the 14th observation lies in the 25 class (midpoint 25).
So, Median class = 25.
Step 3: Apply the Median formula
Where:
• L = Lower boundary of the median class
• N = Total frequency
• CF = Cumulative frequency before the median class
• f = Frequency of median class
• h = Class size (width of interval)
Step 4: Find values for the formula
Here, the class intervals are not directly given, but we can assume them from the midpoints.
For example:
• Midpoint 5 → Class interval (0–10)
• Midpoint 15 → Class interval (10–20)
• Midpoint 25 → Class interval (20–30)
• Midpoint 35 → Class interval (30–40)
• Midpoint 45 → Class interval (40–50)
So, Median class = 20–30.
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Now:
Step 5: Substitute values
So, the median = 22.5.
Detective case two solved!
Wrapping It All Together
• In Part (i), we learned that a wrong entry in data can change the mean. By adjusting
the sum, we found the correct mean = 40.18.
• In Part (ii), we explored how to find the median in grouped frequency data. By
carefully identifying the median class and applying the formula, we got the median =
22.5.
Another Little Story to Remember
Think of a cricket team’s scorecard. The coach wants to know:
• Mean runs scored: This tells him how much, on average, each player scores. But if
someone mistakenly noted a player’s score wrong, the average gets misleading—just
like our first problem.
• Median runs scored: This tells him the middle player’s score when all are arranged in
order. It balances the lower half and the upper half—just like our second problem.
So, both mean and median are important, but they answer different questions.
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Final Answer
1. Correct Mean = 40.18
2. Median = 22.5
(i) What do you mean by Standard Deviation? Write its merits and demerits. 5
(ii) From the following observations, calculate quartile deviation and coefficient of quartile
deviation:
59, 62, 65, 64, 63, 61, 60, 56, 58, 66. 5
Ans:(i) Standard Deviation — what it really means (told like a simple story)
A different beginning
Imagine a school assembly where every student is asked to stand in a straight line by height.
The principal doesn’t just want to know the average height of the class—she wants to know
how tightly or loosely the students are clustered around that average. Are most of them
close to the average, or are there some very short and very tall students that make the line
look “spread out”?
That “spread-outness” is exactly what Standard Deviation (SD) measures.
Simple idea (no jargon first)
• Mean tells you the “center” of the data.
• Standard Deviation tells you how far, on average, data values tend to wander away
from the mean.
If the heights are almost the same, SD is small (everyone huddles near the center). If the
heights vary a lot, SD is large (people spread out far from the center).
A tiny story
Riya and Kabir both run tuition batches of 10 students each. In Riya’s batch, most test scores
hover around 70—some 68, 71, 72—very tight. In Kabir’s batch, scores bounce from 40 to
95—wild spread. Both batches might have the same average (say 70), but Riya’s set will
have a low SD (scores close together), while Kabir’s set has a high SD (scores scattered). This
is why SD matters—two groups with the same average can behave very differently.
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Merits (why SD is so loved)
1. Uses every observation.
SD is computed from all values, not just a few. So it captures the entire dataset’s
variability.
2. Mathematically elegant.
It plays nicely with algebra, calculus, and probability theory. Many statistical tests (z-
scores, t-tests, regression, control charts) rely directly on SD.
3. Units are meaningful.
Since we take the square root of variance, SD comes back to the original units (cm,
marks, rupees), making interpretation intuitive.
4. Works well around the mean.
If your mean is a good “center,” SD complements it perfectly—together they
summarize central tendency and dispersion.
5. Stable across transformations.
If you add a constant to all data, SD stays the same. If you multiply data by a
constant, SD scales by the absolute value of that constant—predictable behavior
that’s useful in practice.
6. Foundational for normal distribution.
In bell-curve situations, the famous 68–95–99.7 rule uses SD: about 68% of data lie
within 1 SD of the mean, ~95% within 2 SDs, etc. (when the data are roughly
normal).
7. Great for comparing variability across similar datasets.
With the same units and similar means, SD makes comparison straightforward.
Demerits (where SD can disappoint)
1. Sensitive to outliers.
Because deviations are squared, a single extreme value can inflate SD a lot. If your
data have outliers, SD might exaggerate overall variability.
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2. Mean-dependent.
SD is built around deviations from the mean. If the mean isn’t an appropriate center
(e.g., heavily skewed data), SD may be misleading.
3. Not robust for skewed distributions.
In skewed datasets, median-based measures (like quartile deviation) may describe
spread more honestly.
4. Less intuitive without the mean.
SD is meaningful mainly in tandem with the mean; by itself, it doesn’t tell a complete
story.
5. Harder to compute by hand.
Compared with range or quartile deviation, SD’s formula is more involved (though
calculators/software make this easy).
6. Comparisons across very different scales are tricky.
If datasets have different units or very different means, raw SDs aren’t directly
comparable—you’d want coefficient of variation instead.
(ii). Quartile Deviation & its Coefficient — step by step on your data
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Bringing it together: when to use which measure?
• Use Standard Deviation when:
o You want a mathematically powerful, mean-based measure of spread.
o Data are reasonably symmetric or bell-shaped.
o You plan to use inferential statistics (z-scores, t-tests, regression).
• Use Quartile Deviation when:
o You want a robust measure less affected by outliers.
o Data are skewed, or you trust the median more than the mean.
o You need a quick, resistant summary (especially in exploratory analysis).
A nice way to think about it: SD is like a zoom lens (precise, detailed, sensitive), and QD is
like sunglasses (cuts glare from extremes to show the general view comfortably).
One last micro-story to make the idea stick
Think of a marching band practicing for Independence Day. The average step length is the
mean. If some players take giant leaps while others shuffle, the line looks messy—that’s
high SD. If almost everyone steps evenly, the line looks neat and tight—that’s low SD. Now
suppose a few players are extremely energetic outliers; the band leader, not wanting those
few to distort the whole picture, checks the quartiles instead—“How spread out are the
central 50%?” That’s where quartile deviation shines: it ignores extreme edges and tells you
how wide the middle corridor really is.
SECTION-C
5. Calculate coefficient of Correlation between X and Y for the following data:
X
1
3
4
5
7
8
10
Y
2
6
8
10
14
16
20
Ans: A Different Beginning
Imagine you are standing in a garden. On one side, there are rows of saplings (small plants)
planted in a line. On the other side, there are pots of water. Every time you give more
water, the plant grows taller. The relationship is simple: more water → more growth.
Now, think for a moment: if you keep records of “amount of water” and “height of plants,”
how can you measure the closeness of this relationship?
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This is exactly what Correlation is. It tells us whether two things are connected, and how
strongly they are connected.
So today, let us walk into this garden of numbers and find out the coefficient of correlation
between X and Y in the given problem. But before solving, let’s build the story a bit further
so it feels natural.
The Story of Rohan and His Marks
There was a student named Rohan. He loved playing cricket, but his parents wanted him to
study. One day, his father made a deal:
• “Rohan, if you study for more hours, your marks will definitely increase. Let’s keep
track of your study hours (X) and your marks (Y).”
Rohan agreed. For a week, he recorded both:
Study Hours (X)
Marks (Y)
1
2
3
6
4
8
5
10
7
14
8
16
10
20
Here, X = hours studied, Y = marks scored.
Now, his father wanted to check: “How strong is the connection between study hours and
marks?”
This strength is measured by Karl Pearson’s Coefficient of Correlation (r).
Step 1: Understanding the Formula
The formula for Karl Pearson’s correlation coefficient is:
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Now make a table:
X
Y
X - X
Y - Ȳ
(X - X)(Y - Ȳ)
(X - X)²
(Y - Ȳ)²
1
2
-4.43
-8.86
39.23
19.63
78.50
3
6
-2.43
-4.86
11.82
5.90
23.62
4
8
-1.43
-2.86
4.09
2.04
8.18
5
10
-0.43
-0.86
0.37
0.18
0.74
7
14
1.57
3.14
4.93
2.47
9.86
8
16
2.57
5.14
13.20
6.61
26.42
10
20
4.57
9.14
41.77
20.90
83.52
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Step 4: Interpretation
Wow! The value of r is almost 1.
This means the relationship between study hours (X) and marks (Y) is nearly perfect positive
correlation.
In simple words: the more Rohan studies, the more marks he gets, almost in exact
proportion.
Connecting Back to Real Life
This result is like saying: if you double the hours of study, marks also almost double. Isn’t
that motivating for students?
But here comes the twist: in real life, things may not be so perfect. Sometimes you study a
lot but the exam is tough, so marks don’t increase as expected. Sometimes you don’t study
much but luckily get easy questions. That’s why correlation in real life is usually less than 1.
Why is Correlation Important?
Let’s pause and think—why do we calculate correlation at all?
• In Business: To check how advertisement expenses (X) affect sales (Y).
• In Agriculture: To see how rainfall (X) affects crop yield (Y).
• In Medicine: To check if exercise hours (X) reduce blood pressure (Y).
• In Education: To know if attendance (X) improves performance (Y).
So correlation is like a bridge of understanding—it connects two variables and tells us how
close they walk together.
A Second Short Story (Engaging Touch)
Think of two friends walking in a park: Aman and Rahul.
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• If Aman walks faster, Rahul also speeds up → Perfect Positive Correlation (+1).
• If Aman walks faster, but Rahul slows down → Perfect Negative Correlation (-1).
• If Aman runs around randomly, Rahul does his own thing → No Correlation (0).
In our problem, X and Y are like best friends walking hand-in-hand. Wherever one goes, the
other follows almost exactly.
Simple Trick to Remember
Correlation is just about togetherness:
• If both rise together → Positive correlation.
• If one rises and other falls → Negative correlation.
• If they don’t care about each other → Zero correlation.
And the coefficient (r) just tells how strong this togetherness is.
Final Answer
The coefficient of correlation between X and Y is:
6. Discuss the importance of time series analysis. Also discuss its components.
Ans: Time Series Analysis: Its Importance and Components
A Fresh Beginning
Imagine you are a farmer. Every year you grow wheat in your fields. Some years your crop is
excellent, some years average, and a few years disappointing. Now you start noticing a
pattern:
• Every summer, because of intense heat, crop yield drops a little.
• During monsoon, crops usually do better.
• Every 4–5 years, heavy floods or droughts disturb production drastically.
Soon you realize—if you keep a record of your harvest year after year, you will be able to
predict future harvests, prepare for bad years, and even decide when to invest in new
farming equipment.
This record of harvest data collected over time is called a time series, and the method of
studying it is time series analysis.
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What is Time Series Analysis?
In simple words, time series analysis means studying data collected at regular time intervals
(daily, monthly, yearly, etc.) to identify patterns, trends, and fluctuations.
• Example: Stock market prices, rainfall records, population growth, sales figures,
temperature readings—all are time series data.
Just like a detective studies clues to solve a mystery, an analyst studies time series data to
forecast the future, understand past behavior, and make informed decisions.
Importance of Time Series Analysis
Time series analysis is not just about numbers; it’s about understanding how things change
over time. Let’s go step by step:
1. Helps in Forecasting
The most powerful use of time series is prediction.
• Businesses forecast sales to plan production.
• Economists forecast inflation and GDP growth.
• Weather departments forecast rainfall and storms.
For example, if a clothing brand knows sales rise every winter, they can stock more jackets
in advance. Without time series analysis, they might end up with too many summer clothes
in December!
In short: Time series helps us look into the future using past data.
2. Guides Business Decisions
Business success depends on correct decisions. Should a company expand production?
Should it reduce prices? Should it invest in new technology?
By analyzing past sales trends, companies can decide the right step. If sales are rising
steadily, expansion makes sense. If sales fluctuate seasonally, businesses can prepare for
“off-seasons.”
In short: Time series is like a GPS for business planning.
3. Measures Economic and Social Changes
Governments and economists rely heavily on time series:
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• Tracking unemployment rates.
• Measuring agricultural production.
• Observing literacy levels over decades.
This helps in policy-making. If unemployment shows an upward trend, the government may
launch new job schemes.
In short: Time series acts like a mirror reflecting society’s growth and challenges.
4. Controls and Improves Operations
Factories use time series to analyze production delays, machine breakdowns, and defects
over time. By identifying patterns, they can reduce waste and improve efficiency.
In short: It turns data into a tool for continuous improvement.
5. Scientific and Technological Applications
Scientists use time series in weather forecasting, earthquake prediction, climate change
studies, and even in space research. For example, studying the time series of global
temperature rise shows the seriousness of global warming.
In short: Time series connects us with nature and science.
A Short Story for Better Understanding
Once upon a time, there was a shopkeeper named Ramesh. He noticed that every year, his
ice-cream sales shot up in May and June and dropped drastically in December and January.
At first, he thought it was just luck.
But when he carefully kept records for 5 years, he realized it was a pattern. So, he started
buying extra stock in summer and reducing stock in winter.
Result? His profits doubled, and he never faced losses due to unsold ice creams again.
This is exactly what time series analysis does: It helps us find hidden patterns and prepare
for the future.
Components of Time Series
A time series is like a dish cooked with different ingredients. To understand it fully, we must
study its four main components:
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1. Trend (T)
• Definition: The long-term direction in which the data moves.
• It may be upward, downward, or stable.
Examples:
• Population generally shows an upward trend.
• Landline usage shows a downward trend.
• Price of gold shows a long-term upward trend.
Why important? Trend tells us the overall picture. Without it, we cannot make long-term
policies.
2. Seasonal Variation (S)
• Definition: Regular changes that repeat every year during the same season or period.
• Usually caused by climate, festivals, or habits.
Examples:
• Ice-cream sales rise in summer.
• Umbrella sales rise in monsoon.
• Sweet sales rise during Diwali or Christmas.
Why important? Businesses can prepare stock and marketing plans for seasonal demand.
3. Cyclical Variation (C)
• Definition: Long-term ups and downs that occur in the economy over a cycle of
several years.
• Usually related to business cycles like boom, recession, depression, and recovery.
Examples:
• Stock markets rise during a boom and fall during recession.
• Real estate prices fluctuate with economic cycles.
Why important? Helps governments and investors prepare for good and bad economic
phases.
4. Irregular or Random Variation (I)
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• Definition: Unpredictable changes caused by unexpected events.
• These cannot be forecasted easily.
Examples:
• Earthquakes, floods, wars, or pandemics (like COVID-19).
• Sudden political decisions affecting markets.
Why important? Even though unpredictable, they remind us to keep safety measures and
emergency plans.
Formula of Time Series
A time series (Y) is generally expressed as:
Y=T+S+C+I
Where:
• T = Trend
• S = Seasonal
• C = Cyclical
• I = Irregular
This simple formula tells us that every time series is a combination of these four forces.
Why Components Matter
Suppose you are analyzing sales of a mobile phone brand.
• Trend shows: sales are increasing because smartphones are becoming essential.
• Seasonal variation shows: sales rise during Diwali or New Year.
• Cyclical variation shows: sales fall during economic slowdown.
• Irregular variation shows: sales drop suddenly due to a supply chain disruption.
By separating these components, analysts can clearly understand what is permanent and
what is temporary.
Another Mini-Story: Weather Department
Think about the weather department. If they simply say “It rained heavily last year, so it will
rain heavily this year too,” it would be foolish.
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Instead, they analyze long-term rainfall trends, seasonal monsoon patterns, cyclical climatic
changes (like El Niño), and unexpected irregular events. Only then can they forecast rainfall
more accurately.
This shows how powerful the study of components is!
Final Words: The Beauty of Time Series
Time series analysis is like a time machine of data. It allows us to travel into the past,
understand the present, and prepare for the future.
• For businesses, it means profit.
• For governments, it means better policies.
• For scientists, it means discoveries.
• For ordinary people, it means planning life better.
And remember, just like our farmer at the beginning of the story, if we carefully observe and
analyze time patterns, we can make smarter decisions in every walk of life.
SECTION-D
5 (i) Three coins are tossed simultaneously. Write its sample space. Also find the
probability of getting at least one head.
(ii) The probability of solving specific problem by A and B is 1/2 and 1/3 respectively. If
both A and B try to solve the problem independently, find the probability that the
problem is solved.
Ans: (i). A little adventure in probability land — coins, friends, and a solved problem
Imagine a sunny afternoon in a small schoolyard. Two scenes are unfolding at once.
Scene 1: Three friends — Maya, Rohan, and Aisha — decide to play a quick game. Each takes
a coin, they all toss their coins at the same time, and they watch them tumble and land on
the ground.
Scene 2: In the classroom across the way, two classmates — Arun and Bhanu — get stuck on
a tricky math problem. Arun is confident he can solve many problems half the time he tries;
Bhanu says she can solve similar problems one time out of three. They both decide to try
independently to solve the problem. What are the chances that at least one of them will
crack it?
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We’ll walk through both scenes like a story, but every step will be clear and mathematical so
any student (and any examiner!) can follow the reasoning and check the arithmetic.
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(ii). Part (ii) — Two people try independently to solve a problem
Now for the classroom story.
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8. (i) What is Binomial Probability distribution? Write its important properties.
(ii) Discuss Normal Probability distribution and its main characteristics.
Ans: Probability Distributions Explained Like a Story
Imagine you are sitting in your classroom. Your teacher walks in and says:
“Today we are going to learn about two very important characters in the world of statistics
— Mr. Binomial Distribution and Mr. Normal Distribution. These two friends appear
everywhere in our daily life, quietly influencing the decisions we make.”
At first, you may think — “Oh, this sounds boring.” But wait! Once you know their story,
you’ll realize they are more like superheroes of mathematics, always helping us solve
problems in business, science, medicine, and even everyday decisions.
Let’s begin with Mr. Binomial Distribution.
Part I: The Story of Binomial Probability Distribution
1. What is Binomial Distribution?
Think of a cricket match. Suppose a batsman faces 10 balls, and for each ball, he either hits a
six (success) or doesn’t (failure). We are interested in knowing:
“What is the probability that he will hit exactly 5 sixes out of 10 balls?”
This type of question can be answered using the Binomial Probability Distribution.
In simple words, Binomial Distribution is used when:
• We have a fixed number of trials (like 10 balls, 20 coin tosses, 50 test samples).
• Each trial has only two possible outcomes → success or failure.
• The probability of success remains the same for every trial.
• All trials are independent (what happens in one trial doesn’t affect the others).
Mathematically, the probability of getting k successes in n trials is given by:
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Part II: The Story of Normal Probability Distribution
Now let’s meet our second character, the most famous one in statistics: Mr. Normal Distribution,
also called the Bell Curve.
1. What is Normal Distribution?
Imagine you and your classmates appear for a math test of 100 marks. When the teacher checks the
copies, not everyone scores the same. Some score very high, some very low, but most students
score somewhere in the middle.
If we plot the marks of all students on a graph, it forms a shape like a bell – highest in the center
(average marks) and tapering off on both sides (very high or very low scores).
That’s exactly what the Normal Probability Distribution looks like.
It is a continuous distribution, unlike the binomial which was discrete.
The mathematical formula of normal distribution looks scary, but its meaning is simple:
Where:
• μ = mean (center of the curve)
• σ = standard deviation (spread of the curve)
But don’t worry about the formula. Just remember: Normal Distribution is a bell-shaped curve that
describes many real-life phenomena.
2. Main Characteristics of Normal Distribution
Let’s list the superpowers of Mr. Normal:
(a) Shape
• It is perfectly symmetrical around the mean.
• Left side is a mirror image of the right side.
(b) Mean, Median, Mode
In a normal distribution:
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They all lie at the center of the curve.
(c) Total Area = 1
The area under the curve represents probabilities, and the total area is always 1 (100%).
(d) Empirical Rule (68-95-99.7 Rule)
This is the most famous property.
• About 68% of data lies within 1 standard deviation from the mean.
• About 95% of data lies within 2 standard deviations.
• About 99.7% of data lies within 3 standard deviations.
Example: If average height of men = 170 cm and standard deviation = 10 cm:
• 68% men will have height between 160 and 180 cm.
• 95% men will have height between 150 and 190 cm.
(e) Continuous
Unlike binomial, the normal distribution is continuous. This means the variable can take any value,
not just integers. For example, someone’s height can be 165.2 cm or 165.8 cm.
(f) Limiting Case
As explained earlier, when the number of trials in a binomial distribution becomes very large, it
approaches a normal distribution. This is why the normal distribution is sometimes called the
limiting form of the binomial.
Short Story to Remember Normal Distribution
Once upon a time, a teacher conducted a surprise test in her class of 50 students. After checking, she
noticed:
• Most students scored around 50 marks.
• A few scored very high (90+) and a few very low (below 20).
When she plotted the marks on a graph, it formed a bell-shaped curve.
She smiled and said:
“This is the magic of normal distribution. Nature, society, and even exam results often follow this
pattern.”
Part III: Comparing Binomial and Normal Distribution
To understand better, let’s compare our two characters:
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Feature
Binomial Distribution
Normal Distribution
Type
Discrete (success/failure)
Continuous (infinite values possible)
Shape
Symmetrical if p = 0.5; skewed otherwise
Always symmetrical bell-shaped
Parameters
n (trials), p (probability of success)
μ (mean), σ (standard deviation)
Mean
np
μ
Variance
np(1-p)
σ²
Connection
For large n, binomial ≈ normal
Limiting case of binomial
Part IV: Why Are These Distributions Important?
• Binomial Distribution helps in:
o Predicting success/failure outcomes (business sales, medical trials, coin tosses,
quality checks).
o Decision-making when events have only two outcomes.
• Normal Distribution helps in:
o Analyzing natural phenomena (heights, weights, intelligence scores).
o Quality control in industries.
o Finance (stock returns often approximate normal distribution).
o Research and scientific studies.
Together, these two form the backbone of probability and statistics.
Conclusion
So, next time you toss a coin, plant a seed, or check your exam marks, remember that two
superheroes – Binomial and Normal Distributions – are silently at work.
• Binomial is the master of “yes/no” type experiments with fixed trials.
• Normal is the universal pattern that governs most continuous data in real life.
They are like two sides of the same coin: one deals with small, countable events, while the other
describes the grand, continuous patterns of nature.
And that’s why statisticians love them so much.
“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”
