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Chapter 3 Probability : Fundamental Concepts 3.1 Set Notation Basic Terms 1. Listing method A = {2, 4, 6, 8, 10} Property-description method A = {x : x is a positive even number not greater than 10} 2. x X x belongs to X or x is an element of X 3. X is the subset of Y. Definition : x X x Y X Y 4. X = Y X and Y are equal sets if and only if X Y and Y X (all elements equal) 5. U universal set 6. null set or empty set 7. Venn Diagrams 8. Finite sets eg. A = {1, 3, 5} 9. Infinite sets eg. T = {t : 5 < t < 10} Operations on Sets 1. Intersection X Y = {x : x X and x Y } X and Y are disjoint if X Y = 2. Union X Y = {x : x X or x Y } 3. Multiple Intersection X1 X2 X3 ... Xn = {x : x belongs to all the X1 ,X2 , ..., Xn} 4. Multiple Union X1 X2 X3 ... Xn = {x : x belongs to at least one of X1 , X2 , ... , Xn } 5. Complement X’ = {x : x U and x X } 6. Relative Complement of X respect to another set Y Y\X = {y : y Y and y X } Fundamental Laws in Set Operations 1. The idempotent laws XX=X and XX=X 2. The commutative laws XY=YX and XY=YX 3. The identity laws X=X and X= XU=U and XU=X 4. The complement laws X X’ = U and X X’ = (X’)’ = X U’ = and ’ = U 3.2 Sample space and Events Basic Terms 1. Random Experiments 2. Sample Space - 3. 4. 5. Sample Points Events Mutually Exclusive - 6. Complementary Events - 7. Exhaustive Events - a trial or experiment which will generate a set of known outcomes and the occurrence has a factor of chance. the set of all possible outcomes (denoted by S) Finite sample space eg. S = {1, 2, 3, 4, 5, 6} Infinite sample space eg. S = {t : 5 < t < 10} The elements in the sample space An event is a subset of a sample space The events X1 , X2, ... Xn are said to be mutually exclusive (n 2) if and only if Xi Xj = for i, j = 1,2, ... , n one case of mutually exclusive events because A A’= The events X1 , X2, ... Xn are said to be exhaustive (n 2) if and only if X1 X2 ... Xn = S / U 3.3 Probability IIT_web_material 1. Classical Definition of Probability n ( A) P(A) = n(A) is the number of sample points of event A n(S ) n(S) is the number of sample points of the sample space 2. Relative Frequency Definition of Probability n P(A) = N is the number of trials N n is the number of times that event A occur ** N is very large/ N ** Some Properties of Probability For any event A in S (1) 0 P(A) 1 (2) P(S)=1 (3) If A and B are mutually exclusive events in S. then P(AB) = P(A) + P(B) (4) If A and B are exhaustive events P(AB) = 1 (5) P(impossible events) = 0 (6) P(certain event) = 1 (7) P(A’) = 1 - P(A) 3. 3.4 Methods of Counting 1. 2. Total no No. of objects of objects taken at a time n n n r Remarks No. of permutation all distinct all distinct n! 3. n n k kinds n1 , n2 , n3 ,... nk 4. n n circular permutation IIT_web_material n! (n r )! n! n1 ! n2 !... n k ! (n-1)! Prn