List of Formulas and Statistics Table, Cheat Sheet of Statistics

Download List of Formulas and Statistics Table and more Cheat Sheet Statistics in PDF only on Docsity! List MF20 List of Formulae and Statistical Tables Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795) For use from 2017 in all papers for the above syllabuses. CST317  PURE MATHEMATICS Mensuration Surface area of sphere = 4πr2 Area of curved surface of cone = πr × slant height Trigonometry a2 = b2 + c2 − 2bc cos A Arithmetic series un = a + (n − 1)d Sn = 1 2 n(a + l) = 1 2 n{2a + (n − 1)d} Geometric series un = arn − 1 Sn = (1 ) 1 − − na r r S∞ = 1− a r for r < 1 Summations 12 6 1 ( 1)(2 1) = = + +∑ n r r n n n 13 2 2 4 1 ( 1) = = +∑ n r r n n Binomial series 1 1 1 +      + =     + +      n n n r r r ( ) 1 2 2 ... ... 1 2 − − −      + = + + + + + +            n n n n n r r nn n n a b a a b a b a b b r , (n ∈ ℕ), where ! C !( )!   = =  −  n r n n r r n r ( ) ( ) ( ) ( )21 1 ... 1 1 1 ... ... 1.2 1.2... − − − + + = + + + + +n rn n n n n r x nx x x r ( x < 1, n ∈ ℝ) Logarithms and exponentials ex ln a = ax Complex numbers {r(cos θ + i sin θ)}n = rn(cos nθ + i sin nθ) eiθ = cos θ + i sin θ The roots of zn = 1 are given by z = 2 i e k n π , for k = 0, 1, 2, ..., n – 1 2 Differentiation f(x) f′(x) tan kx k sec2 kx sin−1 x 2 1 1 x− cos−1 x 2 1 1 x − − tan−1 x 2 1 1 x+ sec x sec x tan x cot x − cosec2 x cosec x − cosec x cot x sinh x cosh x cosh x sinh x tanh x sech2 x sinh−1 x 2 1 1 x+ cosh–1 x 2 1 1x − tanh−1 x 2 1 1 x− Integration (+ constant; a > 0 where relevant) f(x) ∫ f( ) dx x sec2 kx 1 k tan kx tan x ln|sec x| cot x ln|sin x| cosec x −ln|cosec x + cot x| = ln|tan( 1 2 x)| sec x ln|sec x + tan x| = ln|tan( 1 2 x + 1 4 π)| sinh x cosh x cosh x sinh x tanh x ln cosh x 2 2 1 a x− sin–1 x a       (|x| < a) 2 2 1 a x+ 1 a tan–1 x a       2 2 1 x a− cosh–1 x a       or ln{x + 2 2 }x a− (x > a) 2 2 1 a x+ sinh–1 x a       or ln{x + 2 2 }x a+ 2 2 1 a x− 1 1ln 2 a x a a x a + = − tanh–1 x a       (|x| < a) 2 2 1 x a− 1 ln 2 x a a x a − + d dd d d d v uu x uv v x x x = −∫ ∫ 5 Area of a sector A = 1 2 2 dr θ∫ (polar coordinates) A = 1 2 d d d d d y xx y t t t  −   ∫ (parametric form) Arc length s = 2d1 d d y x x  +    ∫ (cartesian coordinates) s = 2 2d d d d d x y t t t    +       ∫ (parametric form) s = 2 2 d d d rr θ θ  +    ∫ (polar form) Surface area of revolution 2π dxS y s= ∫ yS =2π dx s∫ Numerical solution of equations The Newton-Raphson iteration for solving f(x) = 0: xn + 1 = xn − f ( ) f ( ) n n x x′ 6 MECHANICS Motion in a circle Transverse velocity: v rθ=  Transverse acceleration: v rθ=   Radial acceleration: 2 2 vr r θ− = − PROBABILITY Probability P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P(A ∩ B) = P(A) P(B | A) P(A | B) = P( )P( ) P( )P( ) P( )P( ) B A A B A A B A A′ ′+ Bayes’ Theorem: P(Aj | B) = P( )P( ) P( )P( ) j j i i A B A A B AΣ Discrete distributions For a discrete random variable X taking values xi with probabilities pi Expectation (mean): E(X) = μ = Σ xi pi Variance: Var(X) = σ2 = Σ(xi − μ)2 pi = Σ 2 ix pi − μ2 For a function g(X) : E(g(X)) = Σ g(xi) pi The probability generating function (P.G.F.) of X is GX (t) = E(tX), and E(X) = G X ′ (1), Var(X) = 2G (1) G (1) {G (1)}X X X ″ ′ ′+ − For Z = X + Y, where X and Y are independent: GZ (t) = GX (t) GY (t) The moment generating function (M.G.F.) of X is MX (t) = E(etX), and E(X) = M X ′ (0), E(X n) = M ( )n X (0), Var(X) = M X ″ (0) − { M X ′ (0)}2 For Z = X + Y, where X and Y are independent: MZ(t) = MX(t) MY(t) Standard discrete distributions Distribution of X P(X = x) Mean Variance P.G.F. M.G.F. Binomial B(n, p) (1 )x n xn p p x −  −    np np(1 – p) (1 – p + pt)n (1 – p + pet)n Poisson Po(λ) e ! x x λ λ− λ λ eλ(t – 1) (e 1)e tλ − Geometric Geo(p) on 1, 2, … p(1 – p)x – 1 1 p 2 1 p p − 1 (1 ) pt p t− − e 1 (1 )e t t p p− − 7 10 CUMULATIVE BINOMIAL PROBABILITIES 0 P( ) (1 ) x n n r r r r X x C p p− = = −∑ø n = 5 p 0.05 0.1 0.15 1/6 0.2 0.25 0.3 1/3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 2/3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.7738 0.5905 0.4437 0.4019 0.3277 0.2373 0.1681 0.1317 0.1160 0.0778 0.0503 0.0313 0.0185 0.0102 0.0053 0.0041 0.0024 0.0010 0.0003 0.0001 0.0001 0.0000 0.0000 1 0.9774 0.9185 0.8352 0.8038 0.7373 0.6328 0.5282 0.4609 0.4284 0.3370 0.2562 0.1875 0.1312 0.0870 0.0540 0.0453 0.0308 0.0156 0.0067 0.0033 0.0022 0.0005 0.0000 2 0.9988 0.9914 0.9734 0.9645 0.9421 0.8965 0.8369 0.7901 0.7648 0.6826 0.5931 0.5000 0.4069 0.3174 0.2352 0.2099 0.1631 0.1035 0.0579 0.0355 0.0266 0.0086 0.0012 3 1.0000 0.9995 0.9978 0.9967 0.9933 0.9844 0.9692 0.9547 0.9460 0.9130 0.8688 0.8125 0.7438 0.6630 0.5716 0.5391 0.4718 0.3672 0.2627 0.1962 0.1648 0.0815 0.0226 4 1.0000 1.0000 0.9999 0.9999 0.9997 0.9990 0.9976 0.9959 0.9947 0.9898 0.9815 0.9688 0.9497 0.9222 0.8840 0.8683 0.8319 0.7627 0.6723 0.5981 0.5563 0.4095 0.2262 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 n = 6 p 0.05 0.1 0.15 1/6 0.2 0.25 0.3 1/3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 2/3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.7351 0.5314 0.3771 0.3349 0.2621 0.1780 0.1176 0.0878 0.0754 0.0467 0.0277 0.0156 0.0083 0.0041 0.0018 0.0014 0.0007 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 1 0.9672 0.8857 0.7765 0.7368 0.6554 0.5339 0.4202 0.3512 0.3191 0.2333 0.1636 0.1094 0.0692 0.0410 0.0223 0.0178 0.0109 0.0046 0.0016 0.0007 0.0004 0.0001 0.0000 2 0.9978 0.9842 0.9527 0.9377 0.9011 0.8306 0.7443 0.6804 0.6471 0.5443 0.4415 0.3438 0.2553 0.1792 0.1174 0.1001 0.0705 0.0376 0.0170 0.0087 0.0059 0.0013 0.0001 3 0.9999 0.9987 0.9941 0.9913 0.9830 0.9624 0.9295 0.8999 0.8826 0.8208 0.7447 0.6563 0.5585 0.4557 0.3529 0.3196 0.2557 0.1694 0.0989 0.0623 0.0473 0.0159 0.0022 4 1.0000 0.9999 0.9996 0.9993 0.9984 0.9954 0.9891 0.9822 0.9777 0.9590 0.9308 0.8906 0.8364 0.7667 0.6809 0.6488 0.5798 0.4661 0.3446 0.2632 0.2235 0.1143 0.0328 5 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9986 0.9982 0.9959 0.9917 0.9844 0.9723 0.9533 0.9246 0.9122 0.8824 0.8220 0.7379 0.6651 0.6229 0.4686 0.2649 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 n = 7 p 0.05 0.1 0.15 1/6 0.2 0.25 0.3 1/3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 2/3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.6983 0.4783 0.3206 0.2791 0.2097 0.1335 0.0824 0.0585 0.0490 0.0280 0.0152 0.0078 0.0037 0.0016 0.0006 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.9556 0.8503 0.7166 0.6698 0.5767 0.4449 0.3294 0.2634 0.2338 0.1586 0.1024 0.0625 0.0357 0.0188 0.0090 0.0069 0.0038 0.0013 0.0004 0.0001 0.0001 0.0000 0.0000 2 0.9962 0.9743 0.9262 0.9042 0.8520 0.7564 0.6471 0.5706 0.5323 0.4199 0.3164 0.2266 0.1529 0.0963 0.0556 0.0453 0.0288 0.0129 0.0047 0.0020 0.0012 0.0002 0.0000 3 0.9998 0.9973 0.9879 0.9824 0.9667 0.9294 0.8740 0.8267 0.8002 0.7102 0.6083 0.5000 0.3917 0.2898 0.1998 0.1733 0.1260 0.0706 0.0333 0.0176 0.0121 0.0027 0.0002 4 1.0000 0.9998 0.9988 0.9980 0.9953 0.9871 0.9712 0.9547 0.9444 0.9037 0.8471 0.7734 0.6836 0.5801 0.4677 0.4294 0.3529 0.2436 0.1480 0.0958 0.0738 0.0257 0.0038 5 1.0000 1.0000 0.9999 0.9999 0.9996 0.9987 0.9962 0.9931 0.9910 0.9812 0.9643 0.9375 0.8976 0.8414 0.7662 0.7366 0.6706 0.5551 0.4233 0.3302 0.2834 0.1497 0.0444 6 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9994 0.9984 0.9963 0.9922 0.9848 0.9720 0.9510 0.9415 0.9176 0.8665 0.7903 0.7209 0.6794 0.5217 0.3017 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 n = 8 p 0.05 0.1 0.15 1/6 0.2 0.25 0.3 1/3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 2/3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.6634 0.4305 0.2725 0.2326 0.1678 0.1001 0.0576 0.0390 0.0319 0.0168 0.0084 0.0039 0.0017 0.0007 0.0002 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.9428 0.8131 0.6572 0.6047 0.5033 0.3671 0.2553 0.1951 0.1691 0.1064 0.0632 0.0352 0.0181 0.0085 0.0036 0.0026 0.0013 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 2 0.9942 0.9619 0.8948 0.8652 0.7969 0.6785 0.5518 0.4682 0.4278 0.3154 0.2201 0.1445 0.0885 0.0498 0.0253 0.0197 0.0113 0.0042 0.0012 0.0004 0.0002 0.0000 0.0000 3 0.9996 0.9950 0.9786 0.9693 0.9437 0.8862 0.8059 0.7414 0.7064 0.5941 0.4770 0.3633 0.2604 0.1737 0.1061 0.0879 0.0580 0.0273 0.0104 0.0046 0.0029 0.0004 0.0000 4 1.0000 0.9996 0.9971 0.9954 0.9896 0.9727 0.9420 0.9121 0.8939 0.8263 0.7396 0.6367 0.5230 0.4059 0.2936 0.2586 0.1941 0.1138 0.0563 0.0307 0.0214 0.0050 0.0004 5 1.0000 1.0000 0.9998 0.9996 0.9988 0.9958 0.9887 0.9803 0.9747 0.9502 0.9115 0.8555 0.7799 0.6846 0.5722 0.5318 0.4482 0.3215 0.2031 0.1348 0.1052 0.0381 0.0058 6 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9987 0.9974 0.9964 0.9915 0.9819 0.9648 0.9368 0.8936 0.8309 0.8049 0.7447 0.6329 0.4967 0.3953 0.3428 0.1869 0.0572 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9998 0.9993 0.9983 0.9961 0.9916 0.9832 0.9681 0.9610 0.9424 0.8999 0.8322 0.7674 0.7275 0.5695 0.3366 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 C U M U L A T IV E B IN O M IA L P R O B A B IL IT IE S 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 06 0. 00 84 0. 07 12 0. 36 98 1. 00 00 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 10 0. 01 15 0. 08 61 0. 40 13 1. 00 00 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 22 0. 01 96 0. 11 84 0. 45 96 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 09 0. 00 83 0. 05 30 0. 22 52 0. 61 26 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 16 0. 01 28 0. 07 02 0. 26 39 0. 65 13 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 05 0. 00 43 0. 02 56 0. 11 09 0. 34 10 0. 71 76 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 06 0. 00 56 0. 03 39 0. 14 09 0. 40 05 0. 76 84 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 14 0. 00 99 0. 05 00 0. 17 98 0. 45 57 0. 80 31 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 07 0. 00 46 0. 02 39 0. 09 22 0. 26 42 0. 55 65 0. 85 78 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 01 0. 00 11 0. 00 90 0. 04 80 0. 17 83 0. 45 73 0. 80 62 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 24 0. 01 55 0. 06 97 0. 22 48 0. 51 55 0. 83 85 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 13 0. 00 79 0. 03 64 0. 12 52 0. 32 26 0. 61 87 0. 88 78 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 03 0. 00 31 0. 01 96 0. 08 56 0. 26 18 0. 56 38 0. 86 58 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 01 0. 00 09 0. 00 64 0. 03 28 0. 12 09 0. 32 22 0. 62 42 0. 89 26 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 06 0. 00 39 0. 01 94 0. 07 26 0. 20 54 0. 44 17 0. 72 51 0. 93 13 1. 00 00 0. 75 0. 00 00 0. 00 01 0. 00 13 0. 01 00 0. 04 89 0. 16 57 0. 39 93 0. 69 97 0. 92 49 1. 00 00 0. 75 0. 00 00 0. 00 00 0. 00 04 0. 00 35 0. 01 97 0. 07 81 0. 22 41 0. 47 44 0. 75 60 0. 94 37 1. 00 00 0. 75 0. 00 00 0. 00 00 0. 00 00 0. 00 04 0. 00 28 0. 01 43 0. 05 44 0. 15 76 0. 35 12 0. 60 93 0. 84 16 0. 96 83 1. 00 00 0. 7 0. 00 00 0. 00 04 0. 00 43 0. 02 53 0. 09 88 0. 27 03 0. 53 72 0. 80 40 0. 95 96 1. 00 00 0. 7 0. 00 00 0. 00 01 0. 00 16 0. 01 06 0. 04 73 0. 15 03 0. 35 04 0. 61 72 0. 85 07 0. 97 18 1. 00 00 0. 7 0. 00 00 0. 00 00 0. 00 02 0. 00 17 0. 00 95 0. 03 86 0. 11 78 0. 27 63 0. 50 75 0. 74 72 0. 91 50 0. 98 62 1. 00 00 2/ 3 0. 00 01 0. 00 10 0. 00 83 0. 04 24 0. 14 48 0. 34 97 0. 62 28 0. 85 69 0. 97 40 1. 00 00 2/ 3 0. 00 00 0. 00 04 0. 00 34 0. 01 97 0. 07 66 0. 21 31 0. 44 07 0. 70 09 0. 89 60 0. 98 27 1. 00 00 2/ 3 0. 00 00 0. 00 00 0. 00 05 0. 00 39 0. 01 88 0. 06 64 0. 17 77 0. 36 85 0. 60 69 0. 81 89 0. 94 60 0. 99 23 1. 00 00 0. 65 0. 00 01 0. 00 14 0. 01 12 0. 05 36 0. 17 17 0. 39 11 0. 66 27 0. 87 89 0. 97 93 1. 00 00 0. 65 0. 00 00 0. 00 05 0. 00 48 0. 02 60 0. 09 49 0. 24 85 0. 48 62 0. 73 84 0. 91 40 0. 98 65 1. 00 00 0. 65 0. 00 00 0. 00 01 0. 00 08 0. 00 56 0. 02 55 0. 08 46 0. 21 27 0. 41 67 0. 65 33 0. 84 87 0. 95 76 0. 99 43 1. 00 00 0. 6 0. 00 03 0. 00 38 0. 02 50 0. 09 94 0. 26 66 0. 51 74 0. 76 82 0. 92 95 0. 98 99 1. 00 00 0. 6 0. 00 01 0. 00 17 0. 01 23 0. 05 48 0. 16 62 0. 36 69 0. 61 77 0. 83 27 0. 95 36 0. 99 40 1. 00 00 0. 6 0. 00 00 0. 00 03 0. 00 28 0. 01 53 0. 05 73 0. 15 82 0. 33 48 0. 56 18 0. 77 47 0. 91 66 0. 98 04 0. 99 78 1. 00 00 0. 55 0. 00 08 0. 00 91 0. 04 98 0. 16 58 0. 37 86 0. 63 86 0. 85 05 0. 96 15 0. 99 54 1. 00 00 0. 55 0. 00 03 0. 00 45 0. 02 74 0. 10 20 0. 26 16 0. 49 56 0. 73 40 0. 90 04 0. 97 67 0. 99 75 1. 00 00 0. 55 0. 00 01 0. 00 11 0. 00 79 0. 03 56 0. 11 17 0. 26 07 0. 47 31 0. 69 56 0. 86 55 0. 95 79 0. 99 17 0. 99 92 1. 00 00 0. 5 0. 00 20 0. 01 95 0. 08 98 0. 25 39 0. 50 00 0. 74 61 0. 91 02 0. 98 05 0. 99 80 1. 00 00 0. 5 0. 00 10 0. 01 07 0. 05 47 0. 17 19 0. 37 70 0. 62 30 0. 82 81 0. 94 53 0. 98 93 0. 99 90 1. 00 00 0. 5 0. 00 02 0. 00 32 0. 01 93 0. 07 30 0. 19 38 0. 38 72 0. 61 28 0. 80 62 0. 92 70 0. 98 07 0. 99 68 0. 99 98 1. 00 00 0. 45 0. 00 46 0. 03 85 0. 14 95 0. 36 14 0. 62 14 0. 83 42 0. 95 02 0. 99 09 0. 99 92 1. 00 00 0. 45 0. 00 25 0. 02 33 0. 09 96 0. 26 60 0. 50 44 0. 73 84 0. 89 80 0. 97 26 0. 99 55 0. 99 97 1. 00 00 0. 45 0. 00 08 0. 00 83 0. 04 21 0. 13 45 0. 30 44 0. 52 69 0. 73 93 0. 88 83 0. 96 44 0. 99 21 0. 99 89 0. 99 99 1. 00 00 0. 4 0. 01 01 0. 07 05 0. 23 18 0. 48 26 0. 73 34 0. 90 06 0. 97 50 0. 99 62 0. 99 97 1. 00 00 0. 4 0. 00 60 0. 04 64 0. 16 73 0. 38 23 0. 63 31 0. 83 38 0. 94 52 0. 98 77 0. 99 83 0. 99 99 1. 00 00 0. 4 0. 00 22 0. 01 96 0. 08 34 0. 22 53 0. 43 82 0. 66 52 0. 84 18 0. 94 27 0. 98 47 0. 99 72 0. 99 97 1. 00 00 1. 00 00 0. 35 0. 02 07 0. 12 11 0. 33 73 0. 60 89 0. 82 83 0. 94 64 0. 98 88 0. 99 86 0. 99 99 1. 00 00 0. 35 0. 01 35 0. 08 60 0. 26 16 0. 51 38 0. 75 15 0. 90 51 0. 97 40 0. 99 52 0. 99 95 1. 00 00 1. 00 00 0. 35 0. 00 57 0. 04 24 0. 15 13 0. 34 67 0. 58 33 0. 78 73 0. 91 54 0. 97 45 0. 99 44 0. 99 92 0. 99 99 1. 00 00 1. 00 00 1/ 3 0. 02 60 0. 14 31 0. 37 72 0. 65 03 0. 85 52 0. 95 76 0. 99 17 0. 99 90 0. 99 99 1. 00 00 1/ 3 0. 01 73 0. 10 40 0. 29 91 0. 55 93 0. 78 69 0. 92 34 0. 98 03 0. 99 66 0. 99 96 1. 00 00 1. 00 00 1/ 3 0. 00 77 0. 05 40 0. 18 11 0. 39 31 0. 63 15 0. 82 23 0. 93 36 0. 98 12 0. 99 61 0. 99 95 1. 00 00 1. 00 00 1. 00 00 0. 3 0. 04 04 0. 19 60 0. 46 28 0. 72 97 0. 90 12 0. 97 47 0. 99 57 0. 99 96 1. 00 00 1. 00 00 0. 3 0. 02 82 0. 14 93 0. 38 28 0. 64 96 0. 84 97 0. 95 27 0. 98 94 0. 99 84 0. 99 99 1. 00 00 1. 00 00 0. 3 0. 01 38 0. 08 50 0. 25 28 0. 49 25 0. 72 37 0. 88 22 0. 96 14 0. 99 05 0. 99 83 0. 99 98 1. 00 00 1. 00 00 1. 00 00 0. 25 0. 07 51 0. 30 03 0. 60 07 0. 83 43 0. 95 11 0. 99 00 0. 99 87 0. 99 99 1. 00 00 1. 00 00 0. 25 0. 05 63 0. 24 40 0. 52 56 0. 77 59 0. 92 19 0. 98 03 0. 99 65 0. 99 96 1. 00 00 1. 00 00 1. 00 00 0. 25 0. 03 17 0. 15 84 0. 39 07 0. 64 88 0. 84 24 0. 94 56 0. 98 57 0. 99 72 0. 99 96 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 13 42 0. 43 62 0. 73 82 0. 91 44 0. 98 04 0. 99 69 0. 99 97 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 10 74 0. 37 58 0. 67 78 0. 87 91 0. 96 72 0. 99 36 0. 99 91 0. 99 99 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 06 87 0. 27 49 0. 55 83 0. 79 46 0. 92 74 0. 98 06 0. 99 61 0. 99 94 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 19 38 0. 54 27 0. 82 17 0. 95 20 0. 99 10 0. 99 89 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 16 15 0. 48 45 0. 77 52 0. 93 03 0. 98 45 0. 99 76 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 11 22 0. 38 13 0. 67 74 0. 87 48 0. 96 36 0. 99 21 0. 99 87 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 23 16 0. 59 95 0. 85 91 0. 96 61 0. 99 44 0. 99 94 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 19 69 0. 54 43 0. 82 02 0. 95 00 0. 99 01 0. 99 86 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 14 22 0. 44 35 0. 73 58 0. 90 78 0. 97 61 0. 99 54 0. 99 93 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 38 74 0. 77 48 0. 94 70 0. 99 17 0. 99 91 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 34 87 0. 73 61 0. 92 98 0. 98 72 0. 99 84 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 28 24 0. 65 90 0. 88 91 0. 97 44 0. 99 57 0. 99 95 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 63 02 0. 92 88 0. 99 16 0. 99 94 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 59 87 0. 91 39 0. 98 85 0. 99 90 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 54 04 0. 88 16 0. 98 04 0. 99 78 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 n = 9 p x = 0 1 2 3 4 5 6 7 8 9 n = 10 p x = 0 1 2 3 4 5 6 7 8 9 10 n = 12 p x = 0 1 2 3 4 5 6 7 8 9 10 11 12 11 C U M U L A T IV E B IN O M IA L P R O B A B IL IT IE S 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 04 0. 00 42 0. 03 01 0. 15 30 0. 51 23 1. 00 00 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 09 0. 00 70 0. 04 29 0. 18 92 0. 55 99 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 15 0. 00 92 0. 04 41 0. 15 84 0. 41 54 0. 77 12 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 05 0. 00 33 0. 01 70 0. 06 84 0. 21 08 0. 48 53 0. 81 47 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 22 0. 01 15 0. 04 67 0. 14 65 0. 35 21 0. 64 33 0. 89 72 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 11 0. 00 56 0. 02 35 0. 07 91 0. 21 01 0. 43 86 0. 71 61 0. 92 57 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 07 0. 00 41 0. 01 91 0. 06 90 0. 19 37 0. 42 05 0. 70 40 0. 92 21 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 04 0. 00 21 0. 01 01 0. 03 78 0. 11 34 0. 27 09 0. 51 32 0. 77 28 0. 94 59 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 04 0. 00 24 0. 01 16 0. 04 39 0. 12 98 0. 30 18 0. 55 19 0. 80 21 0. 95 60 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 15 0. 00 70 0. 02 67 0. 08 17 0. 20 18 0. 40 19 0. 64 82 0. 85 93 0. 97 19 1. 00 00 0. 75 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 22 0. 01 03 0. 03 83 0. 11 17 0. 25 85 0. 47 87 0. 71 89 0. 89 90 0. 98 22 1. 00 00 0. 75 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 16 0. 00 75 0. 02 71 0. 07 96 0. 18 97 0. 36 98 0. 59 50 0. 80 29 0. 93 65 0. 99 00 1. 00 00 0. 7 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 17 0. 00 83 0. 03 15 0. 09 33 0. 21 95 0. 41 58 0. 64 48 0. 83 92 0. 95 25 0. 99 32 1. 00 00 0. 7 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 16 0. 00 71 0. 02 57 0. 07 44 0. 17 53 0. 34 02 0. 55 01 0. 75 41 0. 90 06 0. 97 39 0. 99 67 1. 00 00 2/ 3 0. 00 00 0. 00 00 0. 00 01 0. 00 07 0. 00 40 0. 01 74 0. 05 76 0. 14 95 0. 31 02 0. 52 45 0. 73 88 0. 89 47 0. 97 26 0. 99 66 1. 00 00 2/ 3 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 08 0. 00 40 0. 01 59 0. 05 00 0. 12 65 0. 26 26 0. 45 31 0. 66 09 0. 83 41 0. 94 06 0. 98 63 0. 99 85 1. 00 00 0. 65 0. 00 00 0. 00 00 0. 00 01 0. 00 11 0. 00 60 0. 02 43 0. 07 53 0. 18 36 0. 35 95 0. 57 73 0. 77 95 0. 91 61 0. 97 95 0. 99 76 1. 00 00 0. 65 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 13 0. 00 62 0. 02 29 0. 06 71 0. 15 94 0. 31 19 0. 51 00 0. 71 08 0. 86 61 0. 95 49 0. 99 02 0. 99 90 1. 00 00 0. 6 0. 00 00 0. 00 01 0. 00 06 0. 00 39 0. 01 75 0. 05 83 0. 15 01 0. 30 75 0. 51 41 0. 72 07 0. 87 57 0. 96 02 0. 99 19 0. 99 92 1. 00 00 0. 6 0. 00 00 0. 00 00 0. 00 01 0. 00 09 0. 00 49 0. 01 91 0. 05 83 0. 14 23 0. 28 39 0. 47 28 0. 67 12 0. 83 34 0. 93 49 0. 98 17 0. 99 67 0. 99 97 1. 00 00 0. 55 0. 00 00 0. 00 03 0. 00 22 0. 01 14 0. 04 26 0. 11 89 0. 25 86 0. 45 39 0. 66 27 0. 83 28 0. 93 68 0. 98 30 0. 99 71 0. 99 98 1. 00 00 0. 55 0. 00 00 0. 00 01 0. 00 06 0. 00 35 0. 01 49 0. 04 86 0. 12 41 0. 25 59 0. 43 71 0. 63 40 0. 80 24 0. 91 47 0. 97 19 0. 99 34 0. 99 90 0. 99 99 1. 00 00 0. 5 0. 00 01 0. 00 09 0. 00 65 0. 02 87 0. 08 98 0. 21 20 0. 39 53 0. 60 47 0. 78 80 0. 91 02 0. 97 13 0. 99 35 0. 99 91 0. 99 99 1. 00 00 0. 5 0. 00 00 0. 00 03 0. 00 21 0. 01 06 0. 03 84 0. 10 51 0. 22 72 0. 40 18 0. 59 82 0. 77 28 0. 89 49 0. 96 16 0. 98 94 0. 99 79 0. 99 97 1. 00 00 1. 00 00 0. 45 0. 00 02 0. 00 29 0. 01 70 0. 06 32 0. 16 72 0. 33 73 0. 54 61 0. 74 14 0. 88 11 0. 95 74 0. 98 86 0. 99 78 0. 99 97 1. 00 00 1. 00 00 0. 45 0. 00 01 0. 00 10 0. 00 66 0. 02 81 0. 08 53 0. 19 76 0. 36 60 0. 56 29 0. 74 41 0. 87 59 0. 95 14 0. 98 51 0. 99 65 0. 99 94 0. 99 99 1. 00 00 1. 00 00 0. 4 0. 00 08 0. 00 81 0. 03 98 0. 12 43 0. 27 93 0. 48 59 0. 69 25 0. 84 99 0. 94 17 0. 98 25 0. 99 61 0. 99 94 0. 99 99 1. 00 00 1. 00 00 0. 4 0. 00 03 0. 00 33 0. 01 83 0. 06 51 0. 16 66 0. 32 88 0. 52 72 0. 71 61 0. 85 77 0. 94 17 0. 98 09 0. 99 51 0. 99 91 0. 99 99 1. 00 00 1. 00 00 1. 00 00 0. 35 0. 00 24 0. 02 05 0. 08 39 0. 22 05 0. 42 27 0. 64 05 0. 81 64 0. 92 47 0. 97 57 0. 99 40 0. 99 89 0. 99 99 1. 00 00 1. 00 00 1. 00 00 0. 35 0. 00 10 0. 00 98 0. 04 51 0. 13 39 0. 28 92 0. 49 00 0. 68 81 0. 84 06 0. 93 29 0. 97 71 0. 99 38 0. 99 87 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 3 0. 00 34 0. 02 74 0. 10 53 0. 26 12 0. 47 55 0. 68 98 0. 85 05 0. 94 24 0. 98 26 0. 99 60 0. 99 93 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1/ 3 0. 00 15 0. 01 37 0. 05 94 0. 16 59 0. 33 91 0. 54 69 0. 73 74 0. 87 35 0. 95 00 0. 98 41 0. 99 60 0. 99 92 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 3 0. 00 68 0. 04 75 0. 16 08 0. 35 52 0. 58 42 0. 78 05 0. 90 67 0. 96 85 0. 99 17 0. 99 83 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 3 0. 00 33 0. 02 61 0. 09 94 0. 24 59 0. 44 99 0. 65 98 0. 82 47 0. 92 56 0. 97 43 0. 99 29 0. 99 84 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 25 0. 01 78 0. 10 10 0. 28 11 0. 52 13 0. 74 15 0. 88 83 0. 96 17 0. 98 97 0. 99 78 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 25 0. 01 00 0. 06 35 0. 19 71 0. 40 50 0. 63 02 0. 81 03 0. 92 04 0. 97 29 0. 99 25 0. 99 84 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 04 40 0. 19 79 0. 44 81 0. 69 82 0. 87 02 0. 95 61 0. 98 84 0. 99 76 0. 99 96 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 02 81 0. 14 07 0. 35 18 0. 59 81 0. 79 82 0. 91 83 0. 97 33 0. 99 30 0. 99 85 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 07 79 0. 29 60 0. 57 95 0. 80 63 0. 93 10 0. 98 09 0. 99 59 0. 99 93 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 05 41 0. 22 72 0. 48 68 0. 72 91 0. 88 66 0. 96 22 0. 98 99 0. 99 79 0. 99 96 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 10 28 0. 35 67 0. 64 79 0. 85 35 0. 95 33 0. 98 85 0. 99 78 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 07 43 0. 28 39 0. 56 14 0. 78 99 0. 92 09 0. 97 65 0. 99 44 0. 99 89 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 22 88 0. 58 46 0. 84 16 0. 95 59 0. 99 08 0. 99 85 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 18 53 0. 51 47 0. 78 92 0. 93 16 0. 98 30 0. 99 67 0. 99 95 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 48 77 0. 84 70 0. 96 99 0. 99 58 0. 99 96 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 44 01 0. 81 08 0. 95 71 0. 99 30 0. 99 91 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 n = 14 p x = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n = 16 p x = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12 C U M U L A T IV E B IN O M IA L P R O B A B IL IT IE S 0. 95 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 06 0. 00 33 0. 01 56 0. 06 08 0. 18 78 0. 44 65 0. 78 54 1. 00 00 0. 9 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 05 0. 00 20 0. 00 78 0. 02 58 0. 07 32 0. 17 55 0. 35 26 0. 58 86 0. 81 63 0. 95 76 1. 00 00 0. 85 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 08 0. 00 29 0. 00 97 0. 02 78 0. 06 98 0. 15 26 0. 28 94 0. 47 55 0. 67 83 0. 84 86 0. 95 20 0. 99 24 1. 00 00 5/ 6 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 05 0. 00 20 0. 00 67 0. 01 97 0. 05 06 0. 11 37 0. 22 35 0. 38 36 0. 57 57 0. 76 04 0. 89 72 0. 97 05 0. 99 58 1. 00 00 0. 8 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 02 0. 00 09 0. 00 31 0. 00 95 0. 02 56 0. 06 11 0. 12 87 0. 23 92 0. 39 30 0. 57 25 0. 74 48 0. 87 73 0. 95 58 0. 98 95 0. 99 88 1. 00 00 0. 75 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 02 0. 00 08 0. 00 27 0. 00 82 0. 02 16 0. 05 07 0. 10 57 0. 19 66 0. 32 64 0. 48 57 0. 65 19 0. 79 74 0. 90 21 0. 96 26 0. 98 94 0. 99 80 0. 99 98 1. 00 00 0. 7 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 06 0. 00 21 0. 00 64 0. 01 69 0. 04 01 0. 08 45 0. 15 93 0. 26 96 0. 41 12 0. 56 85 0. 71 86 0. 84 05 0. 92 34 0. 96 98 0. 99 07 0. 99 79 0. 99 97 1. 00 00 1. 00 00 2/ 3 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 07 0. 00 25 0. 00 72 0. 01 88 0. 04 35 0. 08 98 0. 16 60 0. 27 61 0. 41 52 0. 56 83 0. 71 40 0. 83 32 0. 91 62 0. 96 45 0. 98 78 0. 99 67 0. 99 93 0. 99 99 1. 00 00 1. 00 00 0. 65 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 04 0. 00 14 0. 00 45 0. 01 24 0. 03 01 0. 06 52 0. 12 63 0. 21 98 0. 34 52 0. 49 22 0. 64 25 0. 77 53 0. 87 62 0. 94 14 0. 97 67 0. 99 25 0. 99 81 0. 99 97 1. 00 00 1. 00 00 1. 00 00 0. 6 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 09 0. 00 29 0. 00 83 0. 02 12 0. 04 81 0. 09 71 0. 17 54 0. 28 55 0. 42 15 0. 56 89 0. 70 85 0. 82 37 0. 90 60 0. 95 65 0. 98 28 0. 99 43 0. 99 85 0. 99 97 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 55 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 01 0. 00 04 0. 00 16 0. 00 50 0. 01 38 0. 03 34 0. 07 14 0. 13 56 0. 23 09 0. 35 52 0. 49 75 0. 64 08 0. 76 73 0. 86 50 0. 93 06 0. 96 88 0. 98 79 0. 99 60 0. 99 89 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 5 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 07 0. 00 26 0. 00 81 0. 02 14 0. 04 94 0. 10 02 0. 18 08 0. 29 23 0. 42 78 0. 57 22 0. 70 77 0. 81 92 0. 89 98 0. 95 06 0. 97 86 0. 99 19 0. 99 74 0. 99 93 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 45 0. 00 00 0. 00 00 0. 00 00 0. 00 00 0. 00 02 0. 00 11 0. 00 40 0. 01 21 0. 03 12 0. 06 94 0. 13 50 0. 23 27 0. 35 92 0. 50 25 0. 64 48 0. 76 91 0. 86 44 0. 92 86 0. 96 66 0. 98 62 0. 99 50 0. 99 84 0. 99 96 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 4 0. 00 00 0. 00 00 0. 00 00 0. 00 03 0. 00 15 0. 00 57 0. 01 72 0. 04 35 0. 09 40 0. 17 63 0. 29 15 0. 43 11 0. 57 85 0. 71 45 0. 82 46 0. 90 29 0. 95 19 0. 97 88 0. 99 17 0. 99 71 0. 99 91 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 35 0. 00 00 0. 00 00 0. 00 03 0. 00 19 0. 00 75 0. 02 33 0. 05 86 0. 12 38 0. 22 47 0. 35 75 0. 50 78 0. 65 48 0. 78 02 0. 87 37 0. 93 48 0. 96 99 0. 98 76 0. 99 55 0. 99 86 0. 99 96 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 3 0. 00 00 0. 00 01 0. 00 07 0. 00 33 0. 01 22 0. 03 55 0. 08 38 0. 16 68 0. 28 60 0. 43 17 0. 58 48 0. 72 39 0. 83 40 0. 91 02 0. 95 65 0. 98 12 0. 99 28 0. 99 75 0. 99 93 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 3 0. 00 00 0. 00 03 0. 00 21 0. 00 93 0. 03 02 0. 07 66 0. 15 95 0. 28 14 0. 43 15 0. 58 88 0. 73 04 0. 84 07 0. 91 55 0. 95 99 0. 98 31 0. 99 36 0. 99 79 0. 99 94 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 25 0. 00 02 0. 00 20 0. 01 06 0. 03 74 0. 09 79 0. 20 26 0. 34 81 0. 51 43 0. 67 36 0. 80 34 0. 89 43 0. 94 93 0. 97 84 0. 99 18 0. 99 73 0. 99 92 0. 99 98 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 2 0. 00 12 0. 01 05 0. 04 42 0. 12 27 0. 25 52 0. 42 75 0. 60 70 0. 76 08 0. 87 13 0. 93 89 0. 97 44 0. 99 05 0. 99 69 0. 99 91 0. 99 98 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1/ 6 0. 00 42 0. 02 95 0. 10 28 0. 23 96 0. 42 43 0. 61 64 0. 77 65 0. 88 63 0. 94 94 0. 98 03 0. 99 33 0. 99 80 0. 99 95 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 15 0. 00 76 0. 04 80 0. 15 14 0. 32 17 0. 52 45 0. 71 06 0. 84 74 0. 93 02 0. 97 22 0. 99 03 0. 99 71 0. 99 92 0. 99 98 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 1 0. 04 24 0. 18 37 0. 41 14 0. 64 74 0. 82 45 0. 92 68 0. 97 42 0. 99 22 0. 99 80 0. 99 95 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 0. 05 0. 21 46 0. 55 35 0. 81 22 0. 93 92 0. 98 44 0. 99 67 0. 99 94 0. 99 99 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 1. 00 00 n = 30 p x = 0 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 15 CUMULATIVE POISSON PROBABILITIES 0 P( ) e ! x r r X x r λ λ− = = ∑ø λ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 x = 0 0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231 0.9139 1 1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970 0.9962 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 λ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 x = 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725 2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371 3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865 4 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 λ 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 x = 0 0.3679 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 1 0.7358 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 2 0.9197 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 3 0.9810 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 4 0.9963 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 5 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 6 0.9999 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 7 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 λ 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 x = 0 0.1353 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550 1 0.4060 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.2146 2 0.6767 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.4460 3 0.8571 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.6696 4 0.9473 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.8318 5 0.9834 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.9258 6 0.9955 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.9713 7 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.9901 8 0.9998 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.9969 9 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991 10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9998 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 λ 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 x = 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202 1 0.1991 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.0992 2 0.4232 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.2531 3 0.6472 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.4532 4 0.8153 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.6484 5 0.9161 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.8006 6 0.9665 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.8995 7 0.9881 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.9546 8 0.9962 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.9815 9 0.9989 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931 10 0.9997 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.9977 11 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 16 CUMULATIVE POISSON PROBABILITIES λ 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 x = 0 0.0183 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.0074 1 0.0916 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.0439 2 0.2381 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.1333 3 0.4335 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.2793 4 0.6288 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.4582 5 0.7851 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.6335 6 0.8893 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.7767 7 0.9489 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.8769 8 0.9786 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.9382 9 0.9919 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717 10 0.9972 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.9880 11 0.9991 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.9953 12 0.9997 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.9983 13 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 14 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 λ 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 x = 0 0.0067 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001 1 0.0404 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008 2 0.1247 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042 3 0.2650 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149 4 0.4405 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403 5 0.6160 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885 6 0.7622 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.1649 7 0.8666 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.2687 8 0.9319 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918 9 0.9682 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218 10 0.9863 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453 11 0.9945 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520 12 0.9980 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364 13 0.9993 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981 14 0.9998 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400 15 0.9999 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.9665 16 1.0000 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.9823 17 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.9911 18 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957 19 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.9980 20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9991 21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 17