Permutations" Basic principle of counting" Tree diagrams" k = 3, n 1 = 2, n 2 = 3, n 3 = 2. Braille alphabet" 10/31/10

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1 /3/ Basic priciple of coutig" If step ca be performed i ways, step 2 i 2 ways,..., step i ways, the the ordered sequece" (step, step 2,..., step )" ca be performed i 2... ways. " Tree diagrams" = 3, = 2, 2 = 3, 3 = x 2 x 3 = 2 x 3 x 2 + x 2 x 2 = Braille alphabet" I 824, Louis Braille iveted the stadard alphabet for the blid. It uses a six-dot matrix" where some dots are raised. For example, the letter b is" The cofiguratio with o raised dots is useless. Hece there are" 2! 2! 2! 2! 2! 2 - = 63 differet characters. Permutatios" A ordered sequece of out of distict elemets is a permutatio of legth. " = 2, = 3: ab ac ba bc ca cb" Let! " = " 2 " " " ( factorial)" The umber of permutatios of legth out of distict elemets is "! ( )! The umber of permutatios of distict elemets is!! = " 8 8

/3/ Stirligʼs formula" Biomial theorem"!

c, is called a biomial coefficiet" " 5" 25" 5"!" 2".55 25 " 3.4 64 " 2π +/ 2 e 8.2".546 25 " 3.

called a combiatio of out of " The umber of combiatios of out of is choose or" =!!( )!

2 /3/ Stirligʼs formula" Biomial theorem"! 2π +/ 2 e Origially" discovered" by de Moivre" (x+y) 2 =" (x+y) 3 =" (x + y) = c, x y i= c, is called a biomial coefficiet" " 5" 25" 5"!" 2" " " 2π +/ 2 e 8.2" " " Biomial coefficiet" A uordered selectio of out of distict objects is called a combiatio of out of " The umber of combiatios of out of is choose or" =!!( )! Biomial coefficiets" c, = = Defie! =. The" Pascal's triagle" " " 2 " 3 3 " " =!!! =,,, Row cotais" Geerally we have" + =

/3/ Radom umbers" The Mote Carlo method draws digits with equal probability." P(5 cosecutive digits differet) = " The first 8 digits of e = 2.7828... form 6 groups of 5 each.

3 /3/ Radom umbers" The Mote Carlo method draws digits with equal probability." P(5 cosecutive digits differet) = " The first 8 digits of e = form 6 groups of 5 each. If we arrage them ito 6 batches of groups each, the umber of groups with five differet digits are" " for a total of 52 groups of five differet digits, or 52/6 = 32.5%. " Are the digits of e (pseudo-) radom?" Multiomial coefficiets" Let, 2,..., m be itegers with m =." The umber of ways i which a populatio of elemets ca be divided ito m subpopulatios, the first of which has elemets, the secod 2 elemets, etc., is"!! 2!! Bridge hads" 52! = !3!3!3! There are " " " " " " " " " " " differet bridge hads." I how may does each player get a ace?" Capture-recapture" To estimate the umber N of goldfish i the pod at Decatur Elemetary, r = 25 fishes were caught, tagged, ad released. Later a secod sample of = 2 fishes were caught. Amog these = 5 were tagged. How may fishes were there i the pod?" P (catch tagged fishes) = " How ca we figure out N?"

4 /3/ Fridayʼs class" Biomial coefficiets" Multiomial coefficiets" Capture-recapture" Problems" Problem set 3". (a) {(,),(2,),(2,2),(3,),(3,2),(3,3),(4,),(4,2), (4,3),(4,4),(5,),(5,2),(5,3),(5,4),(5,5),(6,),(6,2), (6,3),(6,4),(6,5),(6,6)}" (b) 6/2=.29" (c) All elemets of the form (i,i) have probability /36, all other have 2/36. Total (6+5*2)/36=" (d) Loo at the relative frequecy of these evets ad see if it is closer to the aswer i (b) or i (e)." (e) 9/36=.25" 2. Let people be ABC ad their hats abc." P(oe get their ow)=p(a gets b)p(b gets c A gets b)+p(a gets c)p(b gets a A gets c) = /3x/2 + /3x/2 = /3." For the geeral case, it is the same as the probability that a permutatio of the digits... has oe i their atural positio.let a be the umber of ways this ca be doe. " 9 9 The first idividual/umber has - possible choices. Suppose he pics i. The there are two optios: perso i pics, which reduces the problem to the case -2, or ot, i which case every remaiig idividual has precisely oe forbidde choice, i.e. the case -. Hece a = (-)(a -2 +a - ) with a =, a 2 =. Hece a 3 =2, a 4 =9 etc. The probability will be p =a /!, so p 3 =/3 (as we saw), p 4 =3/8 etc. The geeral formula is " ( ) i i! e i= There are 5 possible choices. The highest umber is 5 i 5 5 of those. Therefore it is 4 i 4 5 of them. Sice {X 5} = {X 4} + {X=5} we get P(X=5) = ( )/ 5 =.2" 4. For =2 we have " P(A A 2 ) = P(A ) + P(A 2 ) P(A A 2 ) P(A ) + P(A 2 ) Hece the iequality is true whe =2. Suppose it holds for =-. The" P( A i ) = P( A i A ) P( A i ) + P(A ) i= i= i= P(A i ) + P(A ) P(A i ) i= i= 93 4

5 /3/ 5. P(Ace 2 ) = P(Ace 2 Ace )P(Ace ) + P(Ace 2 Not Ace )P(Not Ace ) =" 3/5 x 4/52 + 4/5 x 48/52 = 4/52 =" P(Ace )" I fact, drawig the top card has the same probability of gettig a ace as drawig the bottom card or ay card i the (well shuffled) dec" P(C K)P(K) 6. " P(K C) = P(C K c )P(K c ) + P(C K)P(K) p = = ( p) + p m m ( p ) + which is icreasig i m ad p." 94 A 2-year flood" I 2 years there has bee 3 disastrous floodig evet at a flood plai. A embamet project ca be fiished i 3 years. What is the chace of aother disaster before the?" Box with 2 ticets umbered through 2. Draw three ticets, ad let Y = highest umber (worst floodig) o the ticets." P(Y = i) =" P(Y 8) = " 95 Radom variables" A radom variable is a fuctio from the sample space to the reals." Evets ivolvig a radom variable have probabilities that are computed usig" P({s: Y(s) = y } ) = " P(s) i the discrete case." {s:y(s)=y} Istead of writig P({s: Y(s) = y }) we write P(Y=y)." 96 Tay-Sachs disease" Tay-Sachs disease is a rare but fatal geetic disorder affectig maily childre of Jewish or Easter Europea origi. If a couple are both carriers of the disease, their childre each have probability.25 of beig bor with it. If such a couple has four childre, we are iterested i" X = # childre with the disease" What is the sample space?" X(affa) = " P(X = ) =" 97 5

/3/ Weldoʼs dice data" The British statisticia ad biologist W. R. Weldo performed, together with his wife, i the late 9 th cetury a experimet cosistig of 26,36 throws of 2 dice.

6 /3/ Weldoʼs dice data" The British statisticia ad biologist W. R. Weldo performed, together with his wife, i the late 9 th cetury a experimet cosistig of 26,36 throws of 2 dice. " The outcome of oe die was deemed a success if a five or a six occurred. Thus, each trial could have betwee ad 2 successes. If the die is fair, the probability of a success is /3." To get successes we eed a strig of 2 S or F with exactly S. Probability is" P(X = ) = 3 3 Biomial distributio Bi (=2,p=/3)." As it happeed, observed frequecy of S was P(S) = Why?" 98 The biomial distributio" X ~ Bi(,p)" idepedet trials with success probability p" X couts the umber of successes" Li" P(X = ) = K p ( p) 99 The Decatur fish pod" Recall that the probability i a sample of size to catch X = out of the r tagged fishes i a pod with N fishes is" Ways of choosig tagged fishes" P(X = ) = r Ways of choosig N- utagged" N N r The hypergeometric distributio" Drawig a sample of size without replacemet from a populatio of size N, where r elemets are of type A ad w = N-r are of type B, the umber X of type A elemets has distributio " P(X = ) = r N N r r, N r, Number of possible samples" 6

7 /3/ Relatios" What if we draw with replacemet?" If N is large compared to, does it matter whether we draw with or without replacemet?" Modayʼs lecture" Radom variable" Biomial distributio" Hypergeometric distributio" 2 3 Probability mass ad desity fuctios" If a radom variable X oly taes o a discrete set of values (such as the itegers) we call it a discrete radom variable. It ca be described by its probability mass fuctio (pmf)" " "p X (x) = P(X = x) " If the rage of X cotais a iterval, we say that X is a cotiuous radom variable. I this case we describe the radom variable usig its probability desity fuctio (pdf), a (piecewise) cotiuous curve f X (x) such that" P(a < X b) = f X (x)dx b a 4 WARNING!" While i the discrete case " p X (x) = P(X = x), " this is NOT the case i the cotiuous case. " I fact, sice" P(X = x) = we must have P(X = x) =." x x f X (u)du 5 7

8 /3/ The toy collector" A breafast cereal maufacturer puts a toy i each pacage. There are N differet toys, ad they are put i pacages at radom i equal umbers. Let T be the umber of pacages eeded to get at least o of each of the toys." A i = {o type i toy i first pacages}" P(A i ) = " Voter ifluece" The power of a sigle voter i a close presidetial electio is decisive if the state has =2+ voters ad the rest are evely split." P(voter decisive) = " P(A i A j ) = P(A j A j2 A j ) = P(T > ) =" Li" 6 Average power of a voter i a state with voters ad c electoral votes is" "c P(voter decisive)" 7 Waitig i lie" The legth of time, X, that a customer waits i lie for a ba teller is described by" "f X (x) =.2 exp(- x / 5), x >." However, if the customer is ot waited o withi miutes of arrival, he will leave. What is the probability of this evet?" If the customer iteds to visit the ba five times i the ext moth, what is the probability that he will leave the lie at least oce?" The cumulative distributio fuctio" F X (x) = P(X x) (cdf)" P(X > x ) = " P(a < X b ) =" P(X < x ) = " 8 9 8

9 /3/ Fridayʼs lecture" Discrete ad cotiuous radom variables" Probability mass fuctio (pmf)" Probability desity fuctio (pdf)" Cumulative distributio fuctio (cdf)" Problems" Solutios to Problem set 4". There are eighbor pairs out of" " "pairs, so 2/(-) ( 3)." 2 2. First ote that" {X > + } {X > } = {X > + } "so that" P(X > + ) P(X > + X > ) = P(X > ) "Now if q = -p" P(X > ) = p q j = pq q i = q j=+ i= so" P(X > + X > ) = q+ = q = P(X > ) q 3. If the serum has o effect the chace that ay give aimal gets ifected is still.25, ad if aimals get the serum, " X = # {aimals gettig the disease} ~ Bi(,.25)." P(X = ; = ) =.56" P(X ; = 7) =.5" The =7 result would be stroger evidece agaist the serum ot worig." 2 4. If the diagoal were allowed there would be 8! = 4,32 ways of puttig the roos so they caot capture each other. Sice the diagoal is ot allowed, we ca thi of this as the problem of hats, where o oe is allowed to get their ow hat. The aswer is 8 the" ( ) i 8! = 4,833 i! i= 2 5. P(X " 2 = ) = P(X 2 = X = i)p(x = i) i= = =

10 /3/ P(X 2 > ) = 39/64." P(X 2 =,X = ) = ad" = P(X 2 = ) = + (2 ( ) = so" P(X = X 2 = ) = = Let X = #{wis}. I a 3 game series they eed to wi 2, with probability" =.57 I a 7 game series they eed to wi 4, with probability ".55 4 ( ) =.6 (the fifth game is ot played if oe team wo the first four, etc.). They should wat the loger series." 4 Properties of the cdf" Let F(x) be a cdf. The" (a) F is odecreasig" (b) F( ) " = lim F(x) = x (c) F( ) " = 5 Some examples of cumulative distributio fuctios...or...?" Goig from the cdf to the pmf/pdf" Let X be a discrete radom variable. The" "p X (x) = F X (x) F X (x-)" If X is cotiuous, the" f X (x) = d dx F X (x) 6 7

11 /3/ Paretoʼs icome distributio" If X is the aual family icome, " P(X > x) is the proportio earig more tha x." Vilfrido Pareto, Italia ecoomist ( ) foud" that this ca be described as" c x a, x > x >, a >." What is c?" What is the pdf?" Chage of variables" Suppose we ow the distributio of X, but are iterested i the distributio of Y = f(x) where f is a icreasig cotiuous fuctio. The P(Y y) = P(f(X) y) " " " " " "= P(X f - (y))" " " " " "= F X (f - (y))" Does it matter whether X is discrete or cotiuous?" What if f is decreasig?" 8 9 The liear case" Let Y = ax + b where a >. The" F Y (y) = P(aX + b y) = P(X (y-b)/a)." Discrete case" Cotiuous case" a<" 2

Section 5.1 The Basics of Counting

Section 5.1 The Basics of Counting 1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of