What is Probability?
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1 Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t ow what will happe o ay oe experimet, but has a log ru order. The cocept of probability is ecessary i wor with physical biological or social mechaism that geerate observatio that ca ot be predicted with certaity. Example The relative frequecy of such rasom evets with which they occur i a log series of trails is ofte remarably stable. Evets possessig this property are called radom or stochastic evets wee
2 Basic Combiatorics Multiplicatio Priciple Suppose we are to mae a series of decisios. Suppose there are c choices for decisio ad for each of these there are c 2 choices for decisio 2 etc. The the umber of ways the series of decisios ca be made is c c 2 c 3. Example : Suppose I eed to choose a outfit for tomorrow ad I have 2 pairs of jeas to choose from, 3 shirts ad 2 pairs of shoes that matches with this shirts. The I have differet outfits. wee 2
3 Example 2: The Cartesia product of sets A ad B is the set of all pairs (a,b) where a A, b B. If A has 3 elemets (a,a 2,a 3 ) ad B has 2 elemets (b,b 2 ), the their Cartesia product has 6 members; that is A B {(a,b ), (a,b 2 ), (a 2,b ), (a 2,b 2 ), (a 3,b ), (a 3,b 2 )}. Some more exercise:. We toss R differet die, what is the total umber of possible outcome? 2. How may differet digit umbers ca be composed of the digits -7? 3. A questioeer cosists of 5 questios: Geder (f / m), Religio (Christia, Muslim, Jewish, Hidu, others), livig arragemet (residece, shared apartmet, family), spea Frech (yes / o), marital status. I how may possible ways this questioeer ca be aswered? wee 3
4 Permutatio A order arragemet of distict objects is called a permutatio. The umber of ordered arragemets or permutatio of objects is! ( ) ( 2) ( factorial ). By covetio 0!. The umber of ordered arragemets or permutatio of subjects selected from distict objects is ( ) ( 2) ( +). It is also the umber of ordered subsets of size from a set of size. Notatio: P Example: 3 ad 2 ( ) ( 2) ( + )! ( )! The umber of ordered arragemets of subjects selected with replacemet from objects is. wee 4
5 Examples. How may 3 letter words ca be composed from the Eglish Alphabet s.t: (i) No limitatio (ii) The words has 3 differet letters. 2. How may birthday parties ca 0 people have durig a year s.t.: (i) No limitatio (ii) Each birthday is o a differet day people are gettig ito a elevator i a buildig that has 20 floors. (i) I how may ways they ca get off? (ii) I how may ways they ca get off such that each perso gets off o a differet floor? 4. We eed to arrage 4 math boos, 3 physics boos ad oe statistic boo o a shelf. (i) How may possible arragemets exists to do so? (ii) What is the probability that all the math boos will be together? wee 5
6 wee 6 Combiatios The umber of subsets of size from a set of size whe the order does ot matter is deoted by or ( choose ). The umber of uordered subsets of size selected (without replacemet) from available objects is Importat facts: Exercise: Prove the above. C )!!(! 0
7 Example We eed to select 5 committee members form a class of 70 studets. (i) How may possible samples exists? (ii) How may possible samples exists if the committee members all have differet rules? wee 7
8 wee 8 The Biomial Theorem For ay umbers a, b ad ay positive iteger The terms are referred to as biomial coefficiet. ( ) + i i i b a i b a 0
9 Multiomial Coefficiets The umber of ways to partitioig distict objects ito distict groups cotaiig, 2,, objects respectively, where each object appears i exactly oe group ad i is! 2...! 2! It is called the multiomial coefficiets because they occur i the expasio 2 ( a a + + a ) a a + 2 2! 2 i a Where the sum is tae over all i 0,,..., such that i i wee 9
10 Examples. A small compay gives bouses to their employees at the ed of the year. 5 employees are etitled to receive these bouses of whom 7 employees will receive 00$ bous, 3 will receive 000$ bous ad the rest will receive 3000$ bous. I how may possible ways these bouses ca be distributed? 2. We eed to arrage 5 math boos, 4 physics boos ad 2 statistic boo o a shelf. (i) How may possible arragemets exists to do so? (ii) How may possible arragemets exists so that boos of the same subjects will lie side by side? wee 0
What is Probability?
Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability
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