1. When was the first reorded die-shaped objet used? (a 3500 BC (b 1600 BC ( 2 BC (d 1492 AD (e 1904 AD 2. What was it made of? (a stone (b lay ( kryptonite (d bone (e lutefisk 3. How many sides did it have? (a 1 (b 4 ( 6 (d 8 (e 12 4. Where was it disovered? (a England (b China ( Foxwoods Casino (d Egypt (e Babylon 5. Where was the ordinary dek of ards supposed to have been developed? (a England (b China ( U.S.A. (d Egypt (e Babylon 6. When did the study of probability formally begin? (a 2nd C. BC (b 2nd C. AD ( 8th C. AD (d 15th C. AD (e 18th C. AD Sample Spae (S: the set of all possible outomes Event (E: a olletion of outomes Ex: Roll a die S = {1, 2, 3, 4, 5, 6} A: # of pips is odd A = {1, 3, 5} B: # of pips 3 B = {3, 4, 5, 6} C: # of pips is even C = {2, 4, 6} Intersetion ( E F : The set of elements that belong to both E and F Union ( E F : The set of elements that belong to at least one of E or F Complement ( EorE All elements of S that are not in the set E Algebra of Sets Commutative Law EF F E Assoiative Law ( EF GE( F G Distributive Law ( EF G( EG ( F G De Morgan s Laws ( EF ( EF
Classial * vs. Relative Frequeny Interpretations of Probability Classial Definition of Probability Based on equally likely outomes # ways A an our na ( PA ( # points in Sample Spae ns ( Ex: Roll a die S = {1, 2, 3, 4, 5, 6} A: odd # of pips A = {1, 3, 5} B: # of pips 3 B = {3, 4, 5, 6} C: even # of pips C = {2, 4, 6} P(A = P(B = P(A B = P(A C = Axioms of Probability 1. 0 PE ( 1 2. PS ( 1 3. If E1, E2, and E 3 are all mutually exlusive Then, PE ( 1 E E PE 2 3 ( 1 PE ( 2 PE ( 3 Working with the Axioms S E E PS ( PE ( E 1 PE ( PE ( Relative Frequeny Definition of Probability If n(e= # of times E ours in n repetitions of the experiment ne ( Then, PE ( lim n n
Ex: Roll a die S = {1, 2, 3, 4, 5, 6} A: odd # of pips A = {1, 3, 5} B: # of pips 3 B = {3, 4, 5, 6} A B {1,3,4,5,6} P ( A B 5 6 3 4 7 PA ( B PA ( PB (?? 6 6 6 General Addition Rule for Probability 1 3 5 2 4 6 Multipliation Priniple: If E 1 has n 1 outomes and E 2 has n 2 outomes, there are n 1 n 2 possible outomes for the omposite experiment (E 1 and E 2 There are n! permutations of n objets Permutations of n objets taken r at a time: n P r = n (n-1 (n-2 (n-r+1 The # of ways to selet r items from n items where order matters PA ( B Combinations of n objets taken r at a time: n C r = nn ( 1 ( nr1 n! r! ( n r! r! n r The # of ways to selet r items from n items where order DOES NOT matter Distinguishable permutations of n objets with r of one type and (n-r of another
Binomial Coeffiients: Suppose the pound has 8 Golden Retriever puppies and 6 Shepherd puppies (a In how many way an you selet 2 Goldens and 3 Shepherds? (b Answer (a if there are 2 female Shepherds and you an t hoose both females. Pasal s Triangle: row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1
Distinguishable Permutations (ontinued Suppose there are n items with n 1 alike, n 2 alike,, and n s alike. n! There are distinguishable permutations of the n items. n! n! n! 1 2 s n! n When n1n2 n s n, the terms n1! n2! n! n1, n1, n s are alled s Multinomial Coeffiients sine they are the oeffiients in the expansion of the multinomial, ( x1x2 x n s Non-negative Ball and Urn Model Set up: r balls (represented by r 0 s n urns (represented by n-1 s sampling the n urns a total of r times with replaement to put a ball in eah time The # of unordered samples of size r that an be seleted out of n objets with replaement is given by n1 r ( n1 r! (i.e., the # of distinguishable permutations of 0 s & s r r!( n1! The formal statement: Let x1, x2,, x n be non-negative integers with 1 2 x n (representing the # of balls in eah urn There are x x x r n1 r ( n1 r! distint sets of non-negative integers 1 2 r r!( n1! x x x r. satisfying 1 2 n ( x, x,, x n
Ex: (non-negative ball & urn model In how many ways an 6 idential items be distributed among 3 different stores? Conditional Probability Ex: Roll a die S = {1, 2, 3, 4, 5, 6} C: even # of pips C = {2, 4, 6} D: # of pips 4 D = {4, 5, 6} 1 3 5 2 4 6 3 3 PC (, PD ( 6 6 What if we knew that event D ours? What is the probability of C if D ours? If event D ours, then it beomes our redued sample spae: # ways C D an our PC ( D # points in Redued Sample Spae nc ( D nd ( PC ( D PD ( 5 4 6 Definition of Conditional Probability PA ( B If PB ( 0 then PA ( B PB (
Sample Spae (S for rolling 2 die: n(s = 36 (1,1 (1,2 (1,3 (1,4 (1,5 (1,6 (2,1 (2,2 (2,3 (2,4 (2,5 (2,6 (3,1 (3,2 (3,3 (3,4 (3,5 (3,6 (4,1 (4,2 (4,3 (4,4 (4,5 (4,6 (5,1 (5,2 (5,3 (5,4 (5,5 (5,6 (6,1 (6,2 (6,3 (6,4 (6,5 (6,6 What is the probability that the sum on the two die is 6 if they land on different numbers? Event A: the sum is 6 Event B: the die land on different numbers P( A B Events A and B are independent if PA ( B PA ( PB ( Independene for A & B onditional probabilities are the same as unonditional probabilities PA ( B PA ( PB ( PAB ( PA ( PB ( PB ( Example: A ard is drawn at random from an ordinary dek of ards A: event that an Ae ard is drawn C: event that a Club ard is drawn B: event that a Blak ard is drawn Are events A & C independent? Are events B & C independent?