Probability theory is the branch of mathematics concerned with analyzing random phenomena. It studies random variables, stochastic processes, and events that occur randomly or with uncertainty. While individual events are unpredictable, repeating events will exhibit statistical patterns that can be studied and predicted using tools from probability theory like the law of large numbers and central limit theorem. The mathematical foundations of modern probability theory were established by Kolmogorov. Probability theory has applications in fields like risk assessment, decision-making, reliability analysis, and more.

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2. Probability theory is the branch of mathematics concerned
with analysis of random phenomena. The central objects of
probability theory are random variables, stochastic
processes, and events: mathematical abstractions of non-
deterministic events or measured quantities that may either
be single occurrences or evolve over time in an apparently
random fashion. Although an individual coin toss or the roll
of a die is a random event, if repeated many times the
sequence of random events will exhibit certain statistical
patterns, which can be studied and predicted. Two
representative mathematical results describing such
patterns are the law of large numbers and the central limit
theorem.

3. The mathematical theory of probability has its roots in
attempts to analyze games of chance by Gerolamo Cardano in
the sixteenth century, and by Pierre de Fermat and Blaise
Pascal in the seventeenth century (for example the "problem
of points"). Christiaan Huygens published a book on the
subject in 1657.
Initially, probability theory mainly considered discrete
events, and its methods were mainly combinatorial.
Eventually, analytical considerations compelled the
incorporation of continuous variables into the theory.
This culminated in modern probability theory, the
foundations of which were laid by Andrey Nikolaevich
Kolmogorov. Kolmogorov combined the notion of sample
space, introduced by Richard von Mises, and measure theory
and presented his axiom system for probability theory in

4. Like other theories, the theory of probability is a representation of
probabilistic concepts in formal terms—that is, in terms that can be
considered separately from their meaning. These formal terms are
manipulated by the rules of mathematics and logic, and any results are
interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize
probability, namely the Kolmogorov formulation and the Cox formulation.
In Kolmogorov's formulation (see probability space), sets are interpreted
as events and probability itself as a measure on a class of sets. In Cox's
theorem, probability is taken as a primitive (that is, not further
analyzed) and the emphasis is on constructing a consistent assignment of
probability values to propositions. In both cases, the laws of probability
are the same, except for technical details.
There are other methods for quantifying uncertainty, such as the
Dempster-Shafer theory or possibility theory, but those are essentially
different and not compatible with the laws of probability as usually
understood.

5. Probability theory is applied in everyday life in risk assessment and in trade on
commodity markets. Governments typically apply probabilistic methods in
environmental regulation, where it is called pathway analysis. A good example is
the effect of the perceived probability of any widespread Middle East conflict
on oil prices—which have ripple effects in the economy as a whole. An
assessment by a commodity trader that a war is more likely vs. less likely sends
prices up or down, and signals other traders of that opinion. Accordingly, the
probabilities are not assessed independently nor necessarily very rationally.
The theory of behavioral finance emerged to describe the effect of such
groupthink on pricing, on policy, and on peace and conflict.
It can reasonably be said that the discovery of rigorous methods to assess and
combine probability assessments has profoundly affected modern society.
Accordingly, it may be of some importance to most citizens to understand how
odds and probability assessments are made, and how they contribute to
reputations and to decisions, especially in a democracy.
Another significant application of probability theory in everyday life is
reliability. Many consumer products, such as automobiles and consumer
electronics, use reliability theory in product design to reduce the probability of
failure. Failure probability may influence a manufacture's decisions on a
product's warranty.

6. Suppose we throw a dice once. What would be
the
probability of getting a number greater than 4
and lesser than or equal to 4?
--> Let A be an event where a number greater
than 4 is obtained on throwing a dice once.
Number of possible outcomes = 1, 2, 3, 4, 5, 6
=6
Number of possible outcomes for event A =
5, 6 = 2
Let B be an event where a number Hence, than probability of is obtained2/6 = 1/3
lesser the or equal to 4 event A = on
throwing the dice once.
Number of possible outcomes = 1, 2, 3, 4, 5, 6 = 6
Number of possible outcomes for event B = 1, 2, 3, 4 = 4
Hence, the probability of event B = 4/6 = 2/3

7. One card is drawn from a well shuffled pack of 52 cards. Find the probability
of getting a face card.
Let A be an event where a face card is drawn from the pack of 52 cards.
Number of possible outcomes = 52
There are 3 face cards of each type (hearts, clubs, spades, diamonds) in a
well – shuffled pack of cards.
Hence, number of possible outcomes for event A = 3 x 4 = 12
The probability of event A to occu
12/52 = 3/13

8. 20 defective pens are accidentally mixed with 120 good ones. One pen is taken out
at random from this lot. Determine the probability that the pen taken out will
be a defective one.
Let A be an event where the pen randomly taken out will be a
defective one.
Number of possible outcomes = 120
Number of possible outcomes for event A = 120-20 = 100
Hence, the probability of event A to occur = 100/120
= 5/6

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10.  WE would like to express my special thanks of
gratitude to my teacher Rana Sir who gave me the
golden opportunity to do this wonderful project
on the topic Probability, which also helped me in
doing a lot of Research and i came to know about
so many new things. We are really thankful to
them.
 Secondly We would also like to thank my parents
and friends who helped me a lot in finishing this
project within the limited time.
 We are making this project not only for marks but
to also increase my knowledge .
THANKS AGAIN TO ALL WHO HELPED ME.

11. BY
GAURAV NARVANI-27
DILVEER SINGH PAHWA-28
DEEPAK RAICHANDANI-29
& SEAN RODRIGGUES-30
HUSSAIN SUSNERWALA-31