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PROBABILITY

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INTRODUCTION The theory of probability consist of Statistical approach Classical approach Statistical approach It is also known as repeated experiments and observe frequency approach. In this,probability is defined as the ratio of observed to the total frequency.

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Classical approach. In this, probability is defined as the ratio of favourable number of outcomes to the total number of equally likely outcomes(elementary outcomes). DEFICIENCIES & LIMITATIONS These approaches cannot be applied to the experiments which have large no. of outcomes. The classical definition of probability cannot be applied whenever it is not possible to make a simple enumeration of cases which can be considered equally likely.

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The statistical definition has difficulties from a mathematical point of view because an actual limiting number may not really exist. For this reason modern probability theory has been developed axiomatically. This theory of probability was developed by A.N Kolmogrov (1903 -1987)Russian Mathematician in 1933. In order to understand this approach we must know about same basic terms viz.random experiment, elementary events,sample space, compound events etc

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He laid down certain axioms to interpret probability, in his book “Foundation of probability” published in 1933. :KOLMOGROV

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Random experiments The word experiment means an operation which can produce some well defined outcomes. There are two types of experiments viz. (i) Deterministic experiments. (ii) Random or probability experiments Deterministic experiments. The experiments which have a fixed outcome or result no matter any number of times they are repeated. For example: We can definitely say that the sum of measures of angles of a triangle is 180degree.

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RANDOM EXPERIMENT When an experiment is repeated under identical conditions do not produce the outcome every time but the outcome in a trial is one of the several outcomes,then such an experiment is called random experiments. For example: Rolling an unbiased die. Drawing a card from a well shuffled pack of cards.

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Sample space The set of all possible outcomes of a random experiments is called sample space. It is denoted by S. ILLUSTRATION Random experiment of tossing a coin. The possible outcomes of the experiments are H and T. The sample space associated with this experiment is given by : S = {H,T}

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EVENT A subset of sample space associated with random experiments is called a event. For example :the random of throwing a die. S={ 1,2,3,4,5,6} S has 2 6 = 64 subsets.Each one of these 64 subsets is associated with random experiment of through a die.

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Types of events 1. Impossible events & sure events The empty set and sample space S describes the event. In fact empty set is called impossible event and the whole sample space is called sure event E.g. while rolling a die and getting a number divisible by 7.let a be this event. We know that none of the possible outcomes Compound event An event associated with a random experiment is a compound event, if it is the disjoint union of two or more elementary event.

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For e.g. In a single throw of an ordinary die there are six elementary events and the total number of events is 26 =64 so 26 – (6+1)= 57 is the total number of compound events.

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MUTUALLY EXCLUSIVE EVENTS Two or more events are said to be mutually exclusive ( or incompatible ) when only one of the events can occuring a random experiment and two(or more) events cannot simultaneously. If two events associated with a random experiment are mutually exclusive, then the subsets of the sample space representing the two events are disjoint. Illustration ; In tossing a coin, the two events represented by the outcomes “ Head “ and “ Tail “ are mutually exclusive. For, if “head “ turns up, we cannot get “ tail “ and vice versa

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EXHAUSTIVE EVENTS. A set of events is said to be exhaustive if at least one of them occurs in a random experiment. The elementary events in a sample space are exhaustive. All elementary events associated with a random experiment from a set of exhaustive events. Illustration ; Consider the experiment of drawing a card from a well shuffled deck of playing cards. Let A be the event “ card is red “, B be the event “ card is black “ “Clearly, A and Bare exhaustive events

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PROBABILITY THEORYSET NOTATION Sample spaceS Outcome of a random experiment ω Event AA c S Event A has occurred ω €A Event A has not occurred ω€A Event” A or B “AUB

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Supplementary problems 1.A coin is tossed. If it shows head, then a die is thrown.but if it shows tail, then another coin is tossed. Describe the sample space ? Ans- (H1,H2,HH3,H4,H5,H6,THT,TT,). 2.From a bag containing 3 red,2 blue balls two balls are drawn at random. Write down the sample space and then find the set representing the event “ I ball is red and the other blue “.

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AXIOMATIC APPROACH TO PROBABILITY AXIOMATIC APPROACH IS ANOTHER WAY OF DESCRIBING PROBABILITY OF AN EVENT. IN THIS APPROACH SOME AXIOMS OR RULES ARE DEPICTED TO ASSIGN PROBABILITIES. For example, in ‘ a coin tossing ‘ experiment we can assign the number ½ to each of the outcomes H and T. P( H) = ½ and P ( T ) = ½

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Applications Probability theory is applied in everyday life in risk assessment and in trade on financial markets. Governments apply probabilistic methods in environmental regulations 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

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If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of happening or occurrence of A is denoted by P( A ) and is defined as the ratio m/n. Thus, P(A)=m/n

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WORK SHEET 1.One card is drawn from a pack of 52 cards. The probability that it is the card of a king or spade is a) 1/26 b) 3/26 c)4/13 d)3/13 2.Two dice are thrown simultaneously. The probability of obtaining a total score of 5 is a) 1/18 b)1/12 c)1/9 d) none of these 3.A coin is tossed until two heads or a tail is realized. Specify the sample space of the experiment.

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4. A coin is tossed two times in succession.What is the probability that the first toss resulted in a head ? 5. From the pack of 52 cards three cards are drawn at random.Find the probability of drawing exactly two aces. 6. Four coins are tossed simultaneously. What is the probability of getting at least one head? 7.The probability of A,B,C solving a problem are 1/3,2/7and 3/8 respectively.If all the three try to solve the problem simultaneously,find the probability that exactly one of them solve it?

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8. Two cards are drawn from a well shuffled pack of 52cards without replacement. Find the probability that neither a jack nor a card of spade is drawn. 9.The probability that a student will get A,B.,C or D grade are 0.4,0.35,0.15 and 0.1 respectively. Find the probability that she will get a)B or C b) at most C 10.The odds in favour of an event are 3:5. Find the probability of occurrence of this event.

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