CRC
`
`STAN DARD
` PROBA/TITJTEILITY
`STATISTICS TABLES
`AND FORMULAE
`
`STUDENT EDITION
`
`AAAAAAAAAA 20
`
`Complex Innovati oooooooo en
`
`Amgen Ex. 2020
`Complex Innovations v. Amgen
`IPR2016-00085
`
`

`
`CRC
`
`STAN DARD
`PROBAA]T[T)1LITY
`STATISTICS TABLES
`AND FORMULAE
`
`STUDENT EDITION
`
`STEPHEN T<_QKQSKA
`
`DANIEL ZWILLINGER
`
`

`
`Much of this book originally appeared in D. Zwillinger and S. Kokoska, Standard Probability
`and Statistics Tables and Formulae, Chapman & Hall/CRC, 2000. Reprinted courtesy of Chapman
`& Hall[CRC.
`
`Library of Congress Cataloging-in-Publication Data
`
`Catalog. record is available from the Library of Congress.
`
`Thisbook contains information obtained from authentic and highly regarded sources. Reprinted material
`is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
`efforts have been made to publish reliable data and inforrnation,_ but the author and the publisher cannot
`assume responsibility for the validity of all materials or for the consequences of their use.
`
`Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
`or mechanical, including photocopying, microfilming, and recording, or by any information storage or
`retrieval system, without prior permission in writing from the publisher.
`
`The consent of CRC Press LLC does not.extend to copying" for general distribution, for promotion,’ for
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`for such copying.
`
`Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
`
`Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
`used only for identification and explanation, without intent to infringe.
`
`
`
`© 2000 by Chapman & Hall/CRC
`
`No claim to original U.S. Government works
`International Standard Book Number 0-8493-0026-6
`
`Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
`
`Printed on acid—free paper
`
`

`
`CHAPTER 3
`
`Probability
`
`3.1 ALGEBRA OF SETS
`
`A<;_C>f>u
`
`Figure 3.1: Shaded region ::
`A’.
`
`Figure 3.2: Shaded region =
`A U B.
`
`
`
`Figure
`A H B.
`
`Shaded region :
`
`Figure 3.4: Mutually exclu-
`sive sets.
`
`3.2 COIVIBINATORIAL 1\/IETHODVS
`
`In an equally likely outcome experiment, computing the probability of an
`event involves counting. The following techniques are useful for -determining
`the number of outcomes in an event and /or the sample space.
`
`3.2.1 The product rule for ordered pairs
`
`If the first element of an ordered pair can be selected in 111 Ways, and for each
`of these an ways the second element of the pair can be selected in 712 ways,
`then the number of possible pairs is n1 72,2.
`
`19
`
`

`
`20
`
`CHAPTER 3. PROBABILITY
`
`3.2.2 The generalized product rule for k:—tuples
`
`Suppose a sample space,’ or set, consists of ordered collections of k—tup1es.
`If there are n1 choices for the first element, and for each choice of the first
`element there are n2 choices for the second element, .
`.
`.
`, and for each of the
`first kt — 1 elements there are nk choices for the lath element, then there are
`
`’l’L1’TL2 -
`
`-
`
`- nk possible k:—tup1es.
`
`3.2.3 Permutations
`
`The number of permutations of n distinct objects taken k: at a time is
`
`nl
`
`(3.1)
`
`A table of values is on page 210.
`
`3.2.4 Circular permutations
`
`The number of permutations of n distinct objects arranged in a circle is
`(n — 1)!.
`
`3.2.5 I Combinations (binomial coefficients)
`
`is the number of combinations of n distinct ob-
`The binomial coefficient
`jects taken is at a time without regard to order:
`
`C<”=’“> :
`
`I
`
`Z
`
`(3-2)
`
`See page 210 for a table of values. Other formulas involving binomial coeffi-
`cients include:
`
`<2):"<H>si.:n—k+1>:<.z.>
`00> (8) =(Z>
`(Si)
`<c> (Z) = ($1) + (2:1)
`@@WGW“%3””
`<e> (3) — <3‘)
`<—1>"(Z>
`
`Example 3.8: For the 5 element set {a, b, c, d, e} find the number of subsets containing
`exactly 3 elements.
`
`Solution:
`
`(S1) There are
`
`I
`
`=
`
`= 10 subsets containing exactly 3 elements.
`
`

`
`3.2. COMBINATORIAL METHODS
`
`_
`
`21
`
`(S2) The subsets are
`
`(a, b, c)
`(a, d, 6)
`
`(a, b, d)
`(b, c, d)
`
`(a, b, e)
`(b, C, e)
`
`(a, c, d)
`(b, d, e)
`
`(a, C, e)
`(C, d, e)
`
`3.2.6" Sample selection
`
`There are 4 Ways in which a sample of k elements can be obtained from a set
`of 77, distinguishable objects.
`
`Order
`counts?
`No
`Yes
`
`N0
`Yes
`
`Repetitions The sample
`allowed?
`is called a
`No
`k:—combination
`No
`k—permutation
`_
`k—combination
`‘ Yes
`with replacement
`Yes
`k—permutat1on
`with replacement
`
`Number of Ways to
`choose the sample
`002, k)
`P (n,
`R
`0 (n’ k)
`PR (n, k)
`
`_ where
`
`"‘””“> I (Z) Z
`
`:
`
`: nfi —_—
`
`nl '
`’
`
`CHM’ I“)
`
`C(n+k—1,k:)-_-:——————_1)‘
`/<:l('rz — 1):
`
`__
`
`l
`
`PR(n, /6)
`
`II
`
`.717“
`
`Example 3.9: There are 4 ways in which to choose a 2 element sample from the set
`{a;b}=
`
`combination
`permutation
`combination with replacement
`permutation with replacement
`
`C‘(2, 2) = 1
`P(2, 2) = 2
`CflR(2, 2) = 3
`PR(2,2) :: 4
`
`ab
`ab and ba
`aa, ab, and bb
`aa, ab, ba, and bb
`
`3.2.7 Balls into cells
`
`. There are 8 different ways in which 71 balls can be placed into k cells.
`
`

`
`22-
`
`CHAPTER 3. PROBABILITY
`
`Number of ways to
`place n balls into is cells
`k2”
`kl {
`C'(k: + n —— 1, n) = (k+Z—1)
`C(n — 1,19 — 1) =
`
`2
`p1(n) -5- P2
`pk
`
`‘
`
`+ -
`
`-
`
`- + Pk
`
`
`
`Distinguish Distinguish Can cells
`balls?
`cells?
`be empty?
`Yes
`Yes
`Yes
`No
`Yes
`Yes
`Yes
`No
`No
`Yes
`
`No
`No
`No"
`
`No
`Yes
`No
`
`is the Stirling cycle number and pg
`where
`of the number n into exactly is integer pieces.
`
`is the number of partitions
`
`Given n distinguishable balls and k: distinguishable cells, the number of Ways
`in which we can place n1 balls into cell 1, TL2 balls into cell 2,
`.
`.
`.
`, nk balls
`into cell is, is given by the multinomial coefficient (n1 n2’’___ nk) .
`
`3.2.8
`
`l\/Iultinornial coefficients
`
`, nk), is the number
`.
`.
`The multinomial coefficient, (M Mn,” nk) : C'(n; n1 , n2, .
`of Ways of choosing n1 objects, then ’l’L—2 objects, .
`.
`.
`, then nk objects from a
`collection of n distinct -objects Without regard to order. This requires that
`Zj:1 nj : 77,.
`Other vvaysto interpret the multinomial coefficient:
`
`(1) Permutations (all objects not distinct): Given n1 objects of one kind,
`n2 objects of a second kind,
`.
`.
`.
`, nk objects of a kth kind, and n1 +
`n2 + - -- + nk = n. The number of permutations of the 77, objects is
`(TL1 ,n27:..,nk) ‘
`(2) Partitions: The number of ways of partitioning a set of n distinct objects
`intok: subsets with 77,; objects in the first subset, TL2 objects in the second
`subset, .
`.
`.
`, and nk objects in the lath subset is (n1 n2”.__
`The multinomial symbolis numerically evaluated as
`(
`)2
`
`n
`
`n!
`
`'n’17’n'27"'7’n'/€
`
`n1-n2-
`
`M
`
`Example 3.10: The numberof ways to choose 2 objects, then 1 object, t-hen 1 object
`from the set {a, b, c, d} is (2 A: 1) 2: 12; they are as follows (commas separate the ordered
`selections):
`
`{ab, 0, 61}
`{ad, I), c}
`{bd, 0,, c}
`
`{G5, d, 0}
`{ad, c, b}
`{bd, c, a}
`
`{aca 5: C1}
`{(90, a, d}
`{cd, 0,, 1)}
`
`{@027 d, 5}
`{bc, Cl, a}
`{cd, I), a}
`
`

`
`210
`
`13.8.1 Permutations
`
`CHAPTER 13. MISCELLANEOUS TOPICS
`
`This table contains the number of permutations of n distinct things taken m
`at a time, given by (see section 3.2.3):
`
`P(n,m)= :n(n—1>—--(n—m+1)
`
`(13.24)
`
`Permutations
`
`1
`
`2
`
`6
`
`7
`
`8
`
`
`
`95040
`
`19958400
`3991680
`665280
`51891840
`8648640
`1235520
`154440
`17297280 121080960
`240240 2162160
`32760 360360 3603600 32432400 259459200
`
`15 210 2730
`
`
`Permutations P(n, m)
`
`n
`9
`
`10
`
`11
`12
`13
`14
`15
`
`7n==
`362880
`
`10
`
`11
`
`12
`
`13
`
`3628800
`
`3628800
`
`19958400
`79833600
`259459200
`726485760
`1816214400
`
`39916800
`239500800
`1037836800
`3632428800
`10897286400
`
`39916800
`479001600
`3113510400
`14529715200
`54486432000
`
`’479001600
`6227020800
`43589145600
`217945728000
`
`6227020800
`87178291200
`653837184000
`
`13.8.2 Combinations
`
`This table contains the number of combinations of 77, distinct things taken m
`at a time, given by (see section 3.2.5):
`
`C(n,m) =
`
`=
`
`(13.25)
`
`
`
`120
`
`
`
`55440
`
`720
`5040
`20160
`60480
`151200
`
`5040
`40320
`181440
`604800
`
`40320
`362880
`1814400
`
`1663200
`
`6652800
`
`720
`2520
`6720
`15120
`30240
`
`
` 332640
`
`
`
`
`2
`
`6
`12
`
`20
`
`30
`42
`56
`72
`90
`
`120
`210
`336
`504
`720
`
`110
`
`132
`
`990
`
`1320
`
`1716
`156
`182 2184
`
`1 2
`
`3
`4
`
`5
`
`6
`7
`8
`‘9
`10
`
`11
`
`12
`
`13
`14
`
`

`
`13.8. SUMS OF POWERS OF INTEGERS
`
`3211
`
`Combinations C’(n, m)
`
`2
`
`3
`
`4
`
`5
`
`6
`
`1
`7
`28
`84
`210
`462
`924
`1716
`3003
`5005
`8008
`12376
`18564
`27132
`38760
`54264
`74613
`100947
`134596
`177100
`230230
`296010
`376740
`475020
`593775
`736281
`906192
`1107568
`1344904
`1623160
`1947792
`2324784
`2760681
`3262623
`3838380
`4496388
`5245786 >
`6096454
`7059052
`8145060
`9366819
`10737573
`12271512
`13983816
`15890700
`
`
`1
`8
`366
`120
`330
`792
`1716
`3432
`6435
`11440
`19448
`31824
`50388
`77520
`116280
`170544
`245157
`346104
`480700
`657800
`888030
`1184040
`1560780
`2035800
`2629575
`3365856
`4272048
`5379616
`6724520
`8347680
`10295472
`12620256
`15380937
`18643560
`22481940
`26978328
`32224114
`38320568
`45379620
`‘53524680
`62891499
`73629072
`85900584
`
`
`99884400
`
`
`
`1
`3
`6
`10
`15
`21
`28
`36
`45
`55
`66
`78
`91
`105
`120
`136
`_153
`171
`190
`210
`231
`253
`276
`300
`325
`351
`378
`406
`435
`465
`496
`528
`561
`595
`630
`666
`703
`741
`780
`820
`861
`903
`946
`990
`1035
`1081
`1128
`1176
`1225
`
`1
`4
`10
`20
`35
`56
`84
`120
`165
`220
`286
`364
`455
`560
`680
`816
`969
`1140
`1330
`1540
`1771‘
`2024
`2300
`2600
`2925
`3276
`3654
`4060
`4495
`4960
`5456
`5984
`6545
`7140
`7770
`8436
`9139
`_9880
`10660
`11480
`12341
`.13244
`14190
`15180
`16215
`17296
`18424
`19600
`
`1
`5
`15
`35
`70
`126
`210
`330
`495
`715
`1001
`1365
`1820
`2380
`3060
`3876
`4845
`5985
`7315
`8855 .
`10626
`12650
`14950
`17550
`20475
`23751
`27405
`31465
`35960
`40920
`46376
`52360
`58905
`66045
`73815
`82251
`91390
`101270
`111930
`123410
`135751
`148995
`163185
`178365
`194580
`211876
`230300
`
`1
`6
`21
`56
`126
`252
`462
`792
`1287
`2002
`3003
`4368
`6188
`8568
`11628
`15504
`20349
`26334
`33649
`42504
`53130
`65780
`80730
`98280
`118755
`142506
`169911
`201376
`237336
`278256
`324632
`376992
`435897
`501942
`575757
`658008
`749398
`850668
`962598
`1086008
`1221759
`1370754
`1533939
`1712304
`1906884
`2118760
`
`n
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`11
`12
`13
`14
`15
`16
`17
`18
`19
`20
`21
`22
`23
`24
`25
`26
`27
`28
`29
`30
`31
`32
`33_
`34
`35
`36
`37
`38
`39
`40
`41
`42
`43
`44
`45
`46
`47
`48
`49
`50
`
`7n:::0
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`1
`
`1
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`11
`12
`13
`14
`15
`16'
`17
`18
`19
`20
`321
`22
`23
`24
`25
`26
`27
`28
`29
`30
`31
`32
`33
`34
`35
`36
`37
`38
`39
`40
`41
`42
`43
`44
`45
`46
`47
`48
`49
`50