What is Probability?

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Stanley Goodwin
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1 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 is ecessary i work 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 remarkably stable. Evets possessig this property are called radom or stochastic evets week

2 Relative Frequecy The relative frequecy cocept of probability does ot provide a rigorous defiitio of probability. For our purpose, a iterpretatio based o relative frequecy is a meaigful measure of out belief i the occurrece of the evet. Relative frequecy = proportio of times the eve occurs. Goal: preset a itroductio to the theory of probability, which provides the foudatio for moder statistical iferece. week 2

3 Basic Set Theory A set is a collectio of elemets. Use capital letters, A, B, C to deotes sets ad small letters a, a 2, to deote the elemets of the set. Notatio: meas the elemet a is a elemet of the set A a A A = {a, a 2, a 3 }. The ull, or empty set, deoted by Ф, is the set cosistig of o poits. Thus, Ф is a sub set of every set. The set S cosistig of all elemets uder cosideratio is called the uiversal set. week 3

4 Relatioship Betwee Sets Ay two sets A ad B are equal if A ad B has exactly the same elemets. Notatio: A=B. Example: A = {2, 4, 6}, B = {; is eve ad 2 6} A is a subset of B or A is cotaied i B, if every poit i A is also i B. Notatio: A B Example: A = {2, 4, 6}, B = {; 2 6} = {2, 3, 4, 5, 6} week 4

5 Ve Diagram Sets ad relatioship betwee sets ca be described by usig Ve diagram. Example: We toss a fair die. What is the uiversal set S? week 5

6 Uio ad Itersectio of sets The uio of two sets A ad B, deoted by A B, is the set of all poits that are i at least oe of the sets, i.e., i A or B or both. Example : We toss a fair die The itersectio of two sets A ad B, deoted by A B or AB, is the set of all poits that are members of both A ad B. Example 2: The itersectio of A ad B as defied i example is week 6

7 Properties of uios ad itersectios Uios ad itersectios are: Commutative, i.e., AB = BA ad Associative, i.e., Distributive, i.e., A A A A These laws also apply to arbitrary collectios of sets (ot just pairs). B = ( B C) = ( A B) C B A. ( B C) = ( A B) ( A C) ( B C) = ( A B) ( A C) week 7

8 Disjoit Evets Two sets A ad B are disjoit or mutually exclusive if they have o poits i commo. The A B = Φ. Example 3: Toss a die. Let A = {, 2, 3} ad B = {4, 5}. week 8

9 Complemet of a Set The complemet of a set, deoted by A c or A makes sese oly with respect to some uiversal set. A c is the set of all poits of the uiversal set S that are ot i A. Example: the complemet of set A as defied i example 3 is Note: the sets A ad A c are disjoit. week 9

10 For ay two sets A ad B: C C C ( A B) = A B C C C ( A B) = A B De Morga s Laws For ay collectio of sets A, A 2, A 3, i ay uiversal set S U C A = A = = I( ) C week 0

11 Fiite, Coutable Ifiite ad Ucoutable A set A is fiite if it cotais a fiite umber of elemets. A set A is coutable ifiite if it ca be put ito a oe-to-oe correspodece with the set of positive itegers N. Example: the set of all itegers is coutable ifiite because The whole iterval (0,) is ot coutable ifiite, it is ucoutable. week

12 The Probability Model A experimet is a process by which a observatio is made. For example: roll a die 6 times, toss 3 cois etc. The set of all possible outcomes of a experimet is called the sample space ad is deoted by Ω. The idividual elemets of the sample space are deoted by ω ad are ofte called the sample poits. Examples... A evet is a subset of the sample space. Each sample poit is a simple evet. To defie a probability model we also eed a assessmet of the likelihood of each of these evets. week 2

13 σ Algebra A σ-algebra, F, is a collectio of subsets of Ω satisfyig the followig properties: F cotais Ф ad Ω. F is closed uder takig complemet, i.e., A F F is closed uder takig coutable uio, i.e., A, A2,... F A A2... F A C F. Claim: these properties imply that F is closed uder coutable itersectio. Proof: week 3

14 Probability Measure A probability measure P mappig F [0,] must satisfy A F For, P(A) 0. P(Ω) =. For A A, A,... F, where A i are disjoit, P, 2 3 ( A A A ) = P( A ) + P( A ) + P( ) + L A This property is called coutable additivity. These properties are also called axioms of probability. week 4

15 Formal Defiitio of Probability Model A probability space cosists of three elemets (Ω, F, P) () a set Ω the sample space. (2) a σ-algebra F - collectio of subsets of Ω. (3) a probability measure P mappig F [0,]. week 5

16 Discrete Sample Space A discrete sample space is oe that cotais either a fiite or a coutable umber of distict sample poits. For a discrete sample space it suffices to assig probabilities to each sample poit. There are experimets for which the sample space is ot coutable ad hece is ot discrete. For example, the experimet cosists of measurig the blood pressure of patiets with heart disease. week 6

17 Calculatig Probabilities whe Ω is Fiite Suppose Ω has distict outcomes, Ω = {ω, ω 2,, ω }. The probability of a evet A is P A = P ω ( ) ( ) I may situatios, the outcomes of Ω are equally likely, the, P( ω i ) =. Example, whe rollig a die P i = for i =, 2,, 6. I these situatios the probability that a evet A occurs is () ω A i 6 i P ( A) Example: = # of outcomes for which A occurs = a week 7

18 Basic Combiatorics Multiplicatio Priciple Suppose we are to make 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 = 2 differet outfits. week 8

19 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 )}. Example 3: The umber of subsets of a set of size is 2. E.g. if A = (a,a 2 ) the all the possible subsets of A are: {a }, {a 2 }, {a,a 2 },. Exercises: 2.38, 2.52 i textbook. Some more exercise:. We toss R differet dies, 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), speak Frech (yes / o), marital status. I how may possible ways this questioeer ca be aswered? Φ week 9

20 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 k subjects selected from distict objects is ( ) ( 2) ( k +). It is also the umber of ordered subsets of size k from a set of size. Notatio: P k Example: = 3 ad k = 2 = ( ) ( 2) ( k + ) =! ( k)! The umber of ordered arragemets of k subjects selected with replacemet from objects is k. week 20

21 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 books, 3 physics books ad oe statistic book o a shelf. (i) How may possible arragemets exists to do so? (ii) What is the probability that all the math books will be together? week 2

22 week 22 Combiatios The umber of subsets of size k from a set of size whe the order does ot matter is deoted by or ( choose k ). The umber of uordered subsets of size k selected (without replacemet) from available objects is Importat facts: Exercise: Prove the above. k C k )!!(! k k k = 0 = = = = = k k

23 . Exercise 2.54 from the textbook Examples 2. Exercise 2.55 from the textbook 3. Exercise 2.56 from the textbook 4. 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? week 23

24 week 24 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 k

25 Multiomial Coefficiets The umber of ways to partitioig distict objects ito k distict groups cotaiig, 2,, k objects respectively, k where each object appears i exactly oe group ad i = is! 2... = k! 2! k It is called the multiomial coefficiets because they occur i the expasio 2 ( a a + + a ) = a a k + 2 k 2! k 2 i= a k Where the sum is take over all i = 0,,..., such that k i= i = week 25

26 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 books, 4 physics books ad 2 statistic book o a shelf. (i) How may possible arragemets exists to do so? (ii) How may possible arragemets exists so that books of the same subjects will lie side by side? week 26

What is Probability?

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