1. MY PROJECT ON

2. Probability theory is the branch of mathematics concernedwith analysis of random phenomena. The central objects ofprobability theory are random variables, stochasticprocesses, and events: mathematical abstractions of non-deterministic events or measured quantities that may eitherbe single occurrences or evolve over time in an apparentlyrandom fashion. Although an individual coin toss or the rollof a die is a random event, if repeated many times thesequence of random events will exhibit certain statisticalpatterns, which can be studied and predicted. Tworepresentative mathematical results describing suchpatterns are the law of large numbers and the central limittheorem.

3. The mathematical theory of probability has its roots inattempts to analyze games of chance by Gerolamo Cardano inthe sixteenth century, and by Pierre de Fermat and BlaisePascal in the seventeenth century (for example the "problemof points"). Christiaan Huygens published a book on thesubject in 1657.Initially, probability theory mainly considered discreteevents, and its methods were mainly combinatorial.Eventually, analytical considerations compelled theincorporation of continuous variables into the theory.This culminated in modern probability theory, thefoundations of which were laid by Andrey NikolaevichKolmogorov. Kolmogorov combined the notion of samplespace, introduced by Richard von Mises, and measure theoryand presented his axiom system for probability theory in

4. Like other theories, the theory of probability is a representation ofprobabilistic concepts in formal termsthat is, in terms that can beconsidered separately from their meaning. These formal terms aremanipulated by the rules of mathematics and logic, and any results areinterpreted or translated back into the problem domain.There have been at least two successful attempts to formalizeprobability, namely the Kolmogorov formulation and the Cox formulation.In Kolmogorovs formulation (see probability space), sets are interpretedas events and probability itself as a measure on a class of sets. In Coxstheorem, probability is taken as a primitive (that is, not furtheranalyzed) and the emphasis is on constructing a consistent assignment ofprobability values to propositions. In both cases, the laws of probabilityare the same, except for technical details.There are other methods for quantifying uncertainty, such as theDempster-Shafer theory or possibility theory, but those are essentiallydifferent and not compatible with the laws of probability as usuallyunderstood.

5. Probability theory is applied in everyday life in risk assessment and in trade oncommodity markets. Governments typically apply probabilistic methods inenvironmental regulation, where it is called pathway analysis. A good example isthe effect of the perceived probability of any widespread Middle East conflicton oil priceswhich have ripple effects in the economy as a whole. Anassessment by a commodity trader that a war is more likely vs. less likely sendsprices up or down, and signals other traders of that opinion. Accordingly, theprobabilities are not assessed independently nor necessarily very rationally.The theory of behavioral finance emerged to describe the effect of suchgroupthink on pricing, on policy, and on peace and conflict.It can reasonably be said that the discovery of rigorous methods to assess andcombine probability assessments has profoundly affected modern society.Accordingly, it may be of some importance to most citizens to understand howodds and probability assessments are made, and how they contribute toreputations and to decisions, especially in a democracy.Another significant application of probability theory in everyday life isreliability. Many consumer products, such as automobiles and consumerelectronics, use reliability theory in product design to reduce the probability offailure. Failure probability may influence a manufactures decisions on aproducts 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 = 2Let B be an event where a number Hence, than probability of is obtained2/6 = 1/3 lesser the or equal to 4 event A = onthrowing the dice once.Number of possible outcomes = 1, 2, 3, 4, 5, 6 = 6Number of possible outcomes for event B = 1, 2, 3, 4 = 4Hence, the probability of event B = 4/6 = 2/3

7. One card is drawn from a well shuffled pack of 52 cards. Find the probabilityof 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 = 52There are 3 face cards of each type (hearts, clubs, spades, diamonds) in awell 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 outat random from this lot. Determine the probability that the pen taken out willbe a defective one.Let A be an event where the pen randomly taken out will be adefective one.Number of possible outcomes = 120Number of possible outcomes for event A = 120-20 = 100Hence, 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. BYGAURAV NARVANI-27DILVEER SINGH PAHWA-28DEEPAK RAICHANDANI-29& SEAN RODRIGGUES-30HUSSAIN SUSNERWALA-31