PROBABILITY

A project report submitted to the Department of Mathematics

A project work report by

Name: Abhay Pant
Stream: Science
Grade: XII M12
Roll No:03
NEB Registration Number:
Submitted to: Dr. Santosh Ghimire
The Department of Mathematics
Kathmandu Model Secondary School

Date of Submission:2080-11-18
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 Declaration

I hereby declare that the report of the project work PROBABILTY presented in
this project was done by myself and not submitted elsewhere for examination.
All sources of information have been specially acknowledged by references to
authors or institutions

Signature
Name of the student: Abhay Pant
Stream: Science
Class: 12
Section: M12
Date: 2080-11-18

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 Recommendations

The project work report entitled “PROBABILTY” submitted by Abhay Pant of

Kathmandu Model College, Bag Bazar, Kathmandu.

Is prepared under my supervision as per the procedure and format requirements

laid by the department of Mathematics, Kathmandu Model College, as the partial
fulfillment of requirements of the internal evaluation of Grade XII. Therefore,
recommend the report for evaluation.

…...........................

Signature
Dr. Santosh Ghimire
Date: 2080-11-18

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 Endorsement

We hereby endorse the project report entitled “PROBABILITY” submitted by

Abhay Pant of Kathmandu Model College, Bag bazar, Kathmandu in the partial
fulfillment of requirements of the mathematics subject for internal evaluation.

….................. …...................
Signature Signature
Name of HOD: Mr. Chiranjibi Gyawali Dr. Nagendra Aryal
Department of Mathematics Principal
Date: Date:

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 Acknowledgement

I would like to thank my teacher Dr. Santosh Ghimire who gave me this
opportunity to work on this project. I got to learn a lot from this project about sets.
I would also like to thank our college principal Dr. Nagendra Aryal.

At last, I would like to extend my heartfelt thanks to my parents because without

their help this project would not have been successful.

Abhay Pant

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 Contents

o Declaration
o Acknowledgement
o Recommendation
o Endorsement
o Acknowledgement
o Probability
Experiment
Random Exp.
o Some details about these exp.
Looking at all possible outcomes in various exp.
o Probability of –
Occurrence of event
Examples Ex. 1, 2, 3 and 4
o Conclusion
o Bibliography
o End

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 INTRODUCTION

We have studied Suppose we toss a Again, suppose

the concept of con 500 times and we toss a con
empirical get a head, say, 1000 times and
240 times and tail
probability. get a head, say,
260 times. Then
Since empirical we would say that
530 times and
probability is in a single throw tail 470 times.
based on of a con, the Then, we would
experiments, we probability of say that in a
also call it getting a head as single throw of
experimental 240/500 i.e. 12/25 a coin, the
probability. would say that in a probability of
single throw of a
getting a head is
con, the
probability of
530/1000, i.e.
getting a head as 53/1000
240/500 i.e. 12/25 also call it
we also call it experimental
experimental probability.
probability.

Thus, in various experiments, we would get different probabilities for the same
event
However, theoretical probability overcomes the above problem. In this project by
probability, we shall mean theoretical probability

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 PROBABILITY

Probability is a concept which numerically measures the degree of

certainty of the occurrence of events.
Before defining probability, we shall define certain concepts used there.
Experiment : An operation which can produce more well-defined
outcomes is called an experiment.
Random Experiment : An experiment in which all possible outcomes are
known, and the exact outcome cannot be predicted in advance, is called
a random experiment.
By a trial, we mean ‘performing a random experiment’.
Examples (i) Tossing a fair coin
(ii) Drawing a card from a pack of well-shuffled cards.
These are all examples of a random experiment.
wrote the Provincial Letters in which he defended Jansenism and its
leading philosopher against the Jesuits.

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 SOME DETAILS ABOUT THESE
EXPERIEMENTS

Tossing a coin – When we throw a coin, either a head (H) or a tail (T)
appears on the upper face.
Drawing a card from a well-shuffled deck of 52 cards.
A deck of playing cards has in all 52 cards.
(i) It has 13 cards each of four suits, namely
spades, clubs, hearts and diamonds.
(a) Cards of spades and clubs are black cards.
(b) Cards of hearts and diamonds are red cards.
   
Spades Clubs Hearts Diamonds
(ii) Kings, queens and jacks (or knaves) are known as face
cards. Thus, there are in all 12 face cards.

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 LOOKING AT ALL POSSIBLE
OUTCOMES IN VARIOUS
EXPERIMENTS

When we toss a coin, we get either a head (H) or a tail (T). Thus, all
possible outcomes are H, T :
Suppose two coins are tossed simultaneously. Then, all possible
outcomes are HH, HT, TH, TT
(HH means head on first coin and head on second coin. HT means head
on first coin and tail on second coin etc.)

In drawing a card from a well-shuffled deck of 52 cards, total number of

possible outcomes is 52.

EVENT : The collection of all or some of the possible outcomes is

called an event.

Examples (i) In throwing a coin, H is the event of getting a head, (ii)

Suppose we throw two coins simultaneously and let E be the event of
getting at least one head. Then, E contains HT, TH, HH.

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EQUALLY LIKELY EVENTS A given number of events are said to be
equally likely if none of them is expected to occur in preference to the
others.
For example, if we roll an unbiased die, each number is equally likely to
occur. If, however, a die is so formed that a particular face occurs most
often then the die is biased. In this case, the outcomes are not equally
likely to happen.

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 PROBABILITY OF OCCURRENCE OF AN
EVENT

Probability of occurrence of an event E, denoted by P (E) is defined

as :
P(E) = Number of outcomes favourable to E
Total number of possible outcomes

SURE EVENT - It is evident that in a single toss of die, we will

always get a number less than 7
So, getting a number less than 7 is a sure event.
P (getting a number less than 7) = 6/6 = 1.
Thus, the probability of a sure event is 1.

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IMPOSSIBLE EVENT - In a single toss of a die, what is the
probability of getting a number 8?
We know that in tossing a die, 8 will never come up.
So, getting 8 is an impossible event.
P (getting 8 in a single throw of a die) = 0/6 = 0.
Thus, the probability of an impossible event is zero.

COMPLEMENTARY EVENT – Let E be an event and (not E)

be an event which occurs only when E does not occur.
The event (not E) is called the complementary event of E.
Clearly, P(E) + P (not E) = 1
 P(E) = 1 – P(not E).to the right with the number below
the number to the right.
For example,
 2² → 1+3=4
 3² → 3+6
 4² → 6+10=16

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 SOLVED EXAMPLES

EXAMPLE-1 : A coin is tossed once, what is the probability of getting

a head?

SOLUTION : When a coin is tossed once, all possible outcomes are H

and T.
Total number of possible outcomes = 2.
The favourable outcome is H.
Number of favourable outcomes = 1.
P (getting a head)
= P (H) = number of favourable outcomes = 1
total number of possible outcomes 2

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When we make a pattern by going up and then along, then add up the
values

We will get the Fibonacci sequence.

The Fibonacci Sequence starts "0, 1" and then continues by adding
the two previous numbers, for example 3+5=8, then 5+8=13 etc.

EXAMPLE-2 : A die is thrown once. What is the probability of getting

a prime number?

SOLUTION : In a single throw of a die, all possible outcomes are

1,2,3,4,5,6
Total number of possible outcomes = 6.
Let E be the event of getting a prime number.
Then, the favourable outcomes are 2,3,5
Number of favourable outcomes = 3.
P (getting a prime number) = P(E) = 1 = 1
6 2

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EXAMPLE-3 : A bag contains 5 red balls and some blue balls. If the
probability of drawing a blue ball from the bag is thrice that of a red
ball, find the number of blue balls in the bag.

SOLUTION : Let the number of blue balls in the bag be x.

Then, total number of balls = (5 + x).
Given: P(a blue ball) = 3 x P (a red ball)
x =3X 5  x = 15.
(5+x) (5+x)
Hence, the number of blue balls in the bag is 15.
EXAMPLE-4 : One card is drawn at random from a well-shuffled pack
of 52 cards. What is the probability that the card drawn is either a red
card or a king?

SOLUTION : Total number of all possible outcomes = 52.

Let E be the event of getting a red card or a king.
There are 26 red cards (including 2 kings) and there are 2 more kings.
So, the number of favourable outcomes = (26+2) = 28.
P (getting a red card or a king) = P(E) = 28 = 7
52 13

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 Application Of PROBABILTY in Daily Life

While PROBABILTY is a fundamental mathematical concept, its applications

extend beyond the realm of pure mathematics and find practical use in various
aspects of daily life. Here are a few examples:

Probability and Statistics:

PROBABILTY is closely related to probability distributions and combinations.

The coefficients in the expansion of binomials (which can be read from
PROBABILTY) are used in probability calculations.

In real-life situations, understanding probabilities is crucial, such as in predicting

the likelihood of success in business ventures or the probability of events in risk
analysis.

Genetics:

PROBABILTY is connected to the binomial theorem, which is applied in genetics

to predict the probabilities of different combinations of genes in offspring.

In genetic studies, researchers use probability models derived from PROBABILTY

to analyze and predict the likelihood of specific genetic traits being inherited.

Finance:

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In finance and investment, PROBABILTY can be applied to calculate the
combinations and probabilities associated with different investment scenarios.

Risk assessment and portfolio management often involve probability calculations,

and the binomial coefficients from PROBABILTY can be instrumental in these
computations.

Combinatorial Problems:

PROBABILTY provides a quick method for solving combinatorial problems, such

as counting the number of ways elements can be selected or arranged.

In practical situations, this can be useful in scenarios like optimizing routes for
delivery services or organizing events with different seating arrangements.

Coding and Cryptography:

In computer science, PROBABILTY is used in algorithms for tasks such as error

correction and data compression.

Cryptographic protocols often involve the use of combinatorial principles derived

from PROBABILTY to ensure secure communication and data transmission.

Geometry and Fractals:

PROBABILTY has connections to geometric patterns and fractals. These patterns

are observed in nature and are used in computer graphics and design.

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Architects and artists may find inspiration in the visual symmetry and patterns
derived from PROBABILTY when designing structures or creating aesthetically
pleasing visual compositions.

In essence, PROBABILTY, with its rich mathematical properties, has practical

implications in diverse fields, contributing to problem-solving, decision-making,
and the optimization of various processes in our everyday lives.

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 Conclusion For Maths Project on PROBABILTY

In summary, PROBABILTY stands as a captivating mathematical concept

that has intrigued scholars across generations. Its systematic arrangement of
numbers, governed by straightforward rules, harbors an array of intriguing
patterns and connections. Beyond its apparent simplicity, the
PROBABILTY significance transcends, proving to be a valuable tool in
diverse mathematical and combinatorial contexts.

To sum up, PROBABILTY underscores the beauty and intricacy arising

from seemingly basic principles. Its exploration not only enhances our
comprehension of mathematics but also finds practical applications in areas
such as probability theory and algebraic identities. As we delve into its
complexities, PROBABILTY remains a fount of inspiration for
mathematicians, symbolizing the elegance inherent in the world of numbers
and patterns.

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 BIBLOGRAPHY

 Foundation of Mathematics
Durga Jung K.C
Dr. Santosh Ghimire
Jhavi Lal Ghimire
Ananta Upreti
Amrit Sharma Gautam

 Basic Mathematics
Dr. Ratna Bajracharya
Ram Man Shrestha
Mohan Bir Singh
Yog Ratna Sthapit
Bhanu Chandra Bajracharya

 http://mathforum.org/dr.math/faq/faq.pascal.triangle.html
 https://www.geeksforgeeks.org/pascal-triangle/
 https://medium.com/i-math/top-10-secrets-of-pascals-triangle-6012ba9c5e23
 https://www.mathsisfun.com/pascals-triangle.html