Explicit Expressions for the Exact Number of Simpson Paradox Scenarios By Shalosh B. Ekhad . Definition: For a and b posivie integers, let, A[a, b](n), be the number of Simpson Paradox scenarois with n a, men who took the drug, b n, men who did not take the drug b n, women who took the drug, and , n a, wemen who did not take the drug. ----------------------------------------------------------------------------\ ------- Theorem Number, 1 4 3 35 2 25 A[1, 2](n) = 1/24 n - 5/12 n + -- n - -- n + 1 24 12 and in Maple format A[1,2](n) = 1/24*n^4-5/12*n^3+35/24*n^2-25/12*n+1 hence the asymptotic frequency of a Simpson scenario is 1/96 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 2 ----------------------------------------------------------------------------\ ------- Theorem Number, 2 4 3 2 A[1, 3](n) = 1/6 n - 7/6 n + 17/6 n - 17/6 n + 1 and in Maple format A[1,3](n) = 1/6*n^4-7/6*n^3+17/6*n^2-17/6*n+1 hence the asymptotic frequency of a Simpson scenario is 1/54 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 3 ----------------------------------------------------------------------------\ ------- Theorem Number, 3 4 3 2 A[1, 4](n) = 3/8 n - 9/4 n + 37/8 n - 15/4 n + 1 and in Maple format A[1,4](n) = 3/8*n^4-9/4*n^3+37/8*n^2-15/4*n+1 hence the asymptotic frequency of a Simpson scenario is 3/128 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 4 ----------------------------------------------------------------------------\ ------- Theorem Number, 4 4 3 2 A[1, 5](n) = 2/3 n - 11/3 n + 41/6 n - 29/6 n + 1 and in Maple format A[1,5](n) = 2/3*n^4-11/3*n^3+41/6*n^2-29/6*n+1 hence the asymptotic frequency of a Simpson scenario is 2/75 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 5 ----------------------------------------------------------------------------\ ------- Theorem Number, 5 25 4 65 3 227 2 73 A[1, 6](n) = -- n - -- n + --- n - -- n + 1 24 12 24 12 and in Maple format A[1,6](n) = 25/24*n^4-65/12*n^3+227/24*n^2-73/12*n+1 hence the asymptotic frequency of a Simpson scenario is 25 --- 864 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 6 ----------------------------------------------------------------------------\ ------- Theorem Number, 6 4 3 2 A[1, 7](n) = 3/2 n - 15/2 n + 25/2 n - 15/2 n + 1 and in Maple format A[1,7](n) = 3/2*n^4-15/2*n^3+25/2*n^2-15/2*n+1 hence the asymptotic frequency of a Simpson scenario is 3/98 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7 ----------------------------------------------------------------------------\ ------- Theorem Number, 7 49 4 119 3 383 2 109 A[1, 8](n) = -- n - --- n + --- n - --- n + 1 24 12 24 12 and in Maple format A[1,8](n) = 49/24*n^4-119/12*n^3+383/24*n^2-109/12*n+1 hence the asymptotic frequency of a Simpson scenario is 49 ---- 1536 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 8 ----------------------------------------------------------------------------\ ------- Theorem Number, 8 4 3 2 A[1, 9](n) = 8/3 n - 38/3 n + 119/6 n - 65/6 n + 1 and in Maple format A[1,9](n) = 8/3*n^4-38/3*n^3+119/6*n^2-65/6*n+1 hence the asymptotic frequency of a Simpson scenario is 8/243 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9 ----------------------------------------------------------------------------\ ------- Theorem Number, 9 4 3 2 A[1, 10](n) = 27/8 n - 63/4 n + 193/8 n - 51/4 n + 1 and in Maple format A[1,10](n) = 27/8*n^4-63/4*n^3+193/8*n^2-51/4*n+1 hence the asymptotic frequency of a Simpson scenario is 27 --- 800 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 10 ----------------------------------------------------------------------------\ ------- Theorem Number, 10 4 3 2 A[1, 11](n) = 25/6 n - 115/6 n + 173/6 n - 89/6 n + 1 and in Maple format A[1,11](n) = 25/6*n^4-115/6*n^3+173/6*n^2-89/6*n+1 hence the asymptotic frequency of a Simpson scenario is 25 --- 726 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11 ----------------------------------------------------------------------------\ ------- Theorem Number, 11 4 3 2 A[2, 3](n) = 1/6 n - n + 7/3 n - 5/2 n + 1 and in Maple format A[2,3](n) = 1/6*n^4-n^3+7/3*n^2-5/2*n+1 hence the asymptotic frequency of a Simpson scenario is 1/216 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 3/2 ----------------------------------------------------------------------------\ ------- Theorem Number, 12 4 3 2 A[2, 5](n) = 3/2 n - 5 n + 13/2 n - 4 n + 1 and in Maple format A[2,5](n) = 3/2*n^4-5*n^3+13/2*n^2-4*n+1 hence the asymptotic frequency of a Simpson scenario is 3/200 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 5/2 ----------------------------------------------------------------------------\ ------- Theorem Number, 13 4 3 2 A[2, 7](n) = 25/6 n - 35/3 n + 37/3 n - 35/6 n + 1 and in Maple format A[2,7](n) = 25/6*n^4-35/3*n^3+37/3*n^2-35/6*n+1 hence the asymptotic frequency of a Simpson scenario is 25 ---- 1176 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7/2 ----------------------------------------------------------------------------\ ------- Theorem Number, 14 4 3 2 A[2, 9](n) = 49/6 n - 21 n + 119/6 n - 8 n + 1 and in Maple format A[2,9](n) = 49/6*n^4-21*n^3+119/6*n^2-8*n+1 hence the asymptotic frequency of a Simpson scenario is 49 ---- 1944 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9/2 ----------------------------------------------------------------------------\ ------- Theorem Number, 15 4 3 2 A[2, 11](n) = 27/2 n - 33 n + 29 n - 21/2 n + 1 and in Maple format A[2,11](n) = 27/2*n^4-33*n^3+29*n^2-21/2*n+1 hence the asymptotic frequency of a Simpson scenario is 27 --- 968 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/2 ----------------------------------------------------------------------------\ ------- Theorem Number, 16 4 3 2 A[3, 4](n) = 3/8 n - 7/4 n + 25/8 n - 11/4 n + 1 and in Maple format A[3,4](n) = 3/8*n^4-7/4*n^3+25/8*n^2-11/4*n+1 hence the asymptotic frequency of a Simpson scenario is 1/384 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 4/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 17 4 3 2 A[3, 5](n) = 3/2 n - 9/2 n + 6 n - 4 n + 1 and in Maple format A[3,5](n) = 3/2*n^4-9/2*n^3+6*n^2-4*n+1 hence the asymptotic frequency of a Simpson scenario is 1/150 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 5/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 18 4 3 2 A[3, 7](n) = 6 n - 13 n + 11 n - 5 n + 1 and in Maple format A[3,7](n) = 6*n^4-13*n^3+11*n^2-5*n+1 hence the asymptotic frequency of a Simpson scenario is 2/147 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 19 4 3 2 A[3, 8](n) = 75/8 n - 75/4 n + 129/8 n - 27/4 n + 1 and in Maple format A[3,8](n) = 75/8*n^4-75/4*n^3+129/8*n^2-27/4*n+1 hence the asymptotic frequency of a Simpson scenario is 25 ---- 1536 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 8/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 20 4 3 2 A[3, 10](n) = 147/8 n - 133/4 n + 181/8 n - 31/4 n + 1 and in Maple format A[3,10](n) = 147/8*n^4-133/4*n^3+181/8*n^2-31/4*n+1 hence the asymptotic frequency of a Simpson scenario is 49 ---- 2400 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 10/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 21 4 3 2 A[3, 11](n) = 24 n - 42 n + 30 n - 10 n + 1 and in Maple format A[3,11](n) = 24*n^4-42*n^3+30*n^2-10*n+1 hence the asymptotic frequency of a Simpson scenario is 8/363 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/3 ----------------------------------------------------------------------------\ ------- Theorem Number, 22 4 3 2 A[4, 5](n) = 2/3 n - 8/3 n + 23/6 n - 17/6 n + 1 and in Maple format A[4,5](n) = 2/3*n^4-8/3*n^3+23/6*n^2-17/6*n+1 hence the asymptotic frequency of a Simpson scenario is 1/600 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 5/4 ----------------------------------------------------------------------------\ ------- Theorem Number, 23 4 3 2 A[4, 7](n) = 6 n - 12 n + 12 n - 6 n + 1 and in Maple format A[4,7](n) = 6*n^4-12*n^3+12*n^2-6*n+1 hence the asymptotic frequency of a Simpson scenario is 3/392 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7/4 ----------------------------------------------------------------------------\ ------- Theorem Number, 24 4 3 2 A[4, 9](n) = 50/3 n - 80/3 n + 95/6 n - 35/6 n + 1 and in Maple format A[4,9](n) = 50/3*n^4-80/3*n^3+95/6*n^2-35/6*n+1 hence the asymptotic frequency of a Simpson scenario is 25 ---- 1944 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9/4 ----------------------------------------------------------------------------\ ------- Theorem Number, 25 4 3 2 A[4, 11](n) = 98/3 n - 140/3 n + 94/3 n - 31/3 n + 1 and in Maple format A[4,11](n) = 98/3*n^4-140/3*n^3+94/3*n^2-31/3*n+1 hence the asymptotic frequency of a Simpson scenario is 49 ---- 2904 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/4 ----------------------------------------------------------------------------\ ------- Theorem Number, 26 25 4 3 107 2 A[5, 6](n) = -- n - 15/4 n + --- n - 11/4 n + 1 24 24 and in Maple format A[5,6](n) = 25/24*n^4-15/4*n^3+107/24*n^2-11/4*n+1 hence the asymptotic frequency of a Simpson scenario is 1/864 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 6/5 ----------------------------------------------------------------------------\ ------- Theorem Number, 27 4 3 2 A[5, 7](n) = 25/6 n - 55/6 n + 53/6 n - 29/6 n + 1 and in Maple format A[5,7](n) = 25/6*n^4-55/6*n^3+53/6*n^2-29/6*n+1 hence the asymptotic frequency of a Simpson scenario is 1/294 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7/5 ----------------------------------------------------------------------------\ ------- Theorem Number, 28 4 3 2 A[5, 8](n) = 75/8 n - 65/4 n + 101/8 n - 23/4 n + 1 and in Maple format A[5,8](n) = 75/8*n^4-65/4*n^3+101/8*n^2-23/4*n+1 hence the asymptotic frequency of a Simpson scenario is 3/512 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 8/5 ----------------------------------------------------------------------------\ ------- Theorem Number, 29 4 3 2 A[5, 9](n) = 50/3 n - 25 n + 125/6 n - 17/2 n + 1 and in Maple format A[5,9](n) = 50/3*n^4-25*n^3+125/6*n^2-17/2*n+1 hence the asymptotic frequency of a Simpson scenario is 2/243 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9/5 ----------------------------------------------------------------------------\ ------- Theorem Number, 30 4 3 2 A[5, 11](n) = 75/2 n - 95/2 n + 41/2 n - 13/2 n + 1 and in Maple format A[5,11](n) = 75/2*n^4-95/2*n^3+41/2*n^2-13/2*n+1 hence the asymptotic frequency of a Simpson scenario is 3/242 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/5 ----------------------------------------------------------------------------\ ------- Theorem Number, 31 4 3 2 A[6, 7](n) = 3/2 n - 5 n + 5 n - 5/2 n + 1 and in Maple format A[6,7](n) = 3/2*n^4-5*n^3+5*n^2-5/2*n+1 hence the asymptotic frequency of a Simpson scenario is 1/1176 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 7/6 ----------------------------------------------------------------------------\ ------- Theorem Number, 32 4 3 2 A[6, 11](n) = 75/2 n - 45 n + 33 n - 23/2 n + 1 and in Maple format A[6,11](n) = 75/2*n^4-45*n^3+33*n^2-23/2*n+1 hence the asymptotic frequency of a Simpson scenario is 25 ---- 2904 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/6 ----------------------------------------------------------------------------\ ------- Theorem Number, 33 49 4 77 3 131 2 25 A[7, 8](n) = -- n - -- n + --- n - -- n + 1 24 12 24 12 and in Maple format A[7,8](n) = 49/24*n^4-77/12*n^3+131/24*n^2-25/12*n+1 hence the asymptotic frequency of a Simpson scenario is 1/1536 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 8/7 ----------------------------------------------------------------------------\ ------- Theorem Number, 34 4 3 2 A[7, 9](n) = 49/6 n - 91/6 n + 34/3 n - 16/3 n + 1 and in Maple format A[7,9](n) = 49/6*n^4-91/6*n^3+34/3*n^2-16/3*n+1 hence the asymptotic frequency of a Simpson scenario is 1/486 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9/7 ----------------------------------------------------------------------------\ ------- Theorem Number, 35 4 3 2 A[7, 10](n) = 147/8 n - 105/4 n + 149/8 n - 31/4 n + 1 and in Maple format A[7,10](n) = 147/8*n^4-105/4*n^3+149/8*n^2-31/4*n+1 hence the asymptotic frequency of a Simpson scenario is 3/800 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 10/7 ----------------------------------------------------------------------------\ ------- Theorem Number, 36 4 3 2 A[7, 11](n) = 98/3 n - 119/3 n + 61/3 n - 22/3 n + 1 and in Maple format A[7,11](n) = 98/3*n^4-119/3*n^3+61/3*n^2-22/3*n+1 hence the asymptotic frequency of a Simpson scenario is 2/363 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/7 ----------------------------------------------------------------------------\ ------- Theorem Number, 37 4 3 2 A[8, 9](n) = 8/3 n - 8 n + 35/6 n - 3/2 n + 1 and in Maple format A[8,9](n) = 8/3*n^4-8*n^3+35/6*n^2-3/2*n+1 hence the asymptotic frequency of a Simpson scenario is 1/1944 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 9/8 ----------------------------------------------------------------------------\ ------- Theorem Number, 38 4 3 2 A[8, 11](n) = 24 n - 32 n + 18 n - 7 n + 1 and in Maple format A[8,11](n) = 24*n^4-32*n^3+18*n^2-7*n+1 hence the asymptotic frequency of a Simpson scenario is 3/968 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/8 ----------------------------------------------------------------------------\ ------- Theorem Number, 39 4 3 2 A[9, 10](n) = 27/8 n - 39/4 n + 49/8 n - 3/4 n + 1 and in Maple format A[9,10](n) = 27/8*n^4-39/4*n^3+49/8*n^2-3/4*n+1 hence the asymptotic frequency of a Simpson scenario is 1/2400 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 10/9 ----------------------------------------------------------------------------\ ------- Theorem Number, 40 4 3 2 A[9, 11](n) = 27/2 n - 45/2 n + 27/2 n - 11/2 n + 1 and in Maple format A[9,11](n) = 27/2*n^4-45/2*n^3+27/2*n^2-11/2*n+1 hence the asymptotic frequency of a Simpson scenario is 1/726 confirming that it is ((k-1)/k)^2/24, where k=b/a=, 11/9 ----------------------------------------------------------------------------\ ------- Theorem Number, 41 4 3 2 A[10, 11](n) = 25/6 n - 35/3 n + 19/3 n + 1/6 n + 1 and in Maple format A[10,11](n) = 25/6*n^4-35/3*n^3+19/3*n^2+1/6*n+1 hence the asymptotic frequency of a Simpson scenario is 1/2904 11 confirming that it is ((k-1)/k)^2/24, where k=b/a=, -- 10 To sum up here is the data in Maple format [[[1, 2], 1/24*n^4-5/12*n^3+35/24*n^2-25/12*n+1], [[1, 3], 1/6*n^4-7/6*n^3+17/6 *n^2-17/6*n+1], [[1, 4], 3/8*n^4-9/4*n^3+37/8*n^2-15/4*n+1], [[1, 5], 2/3*n^4-\ 11/3*n^3+41/6*n^2-29/6*n+1], [[1, 6], 25/24*n^4-65/12*n^3+227/24*n^2-73/12*n+1] , [[1, 7], 3/2*n^4-15/2*n^3+25/2*n^2-15/2*n+1], [[1, 8], 49/24*n^4-119/12*n^3+ 383/24*n^2-109/12*n+1], [[1, 9], 8/3*n^4-38/3*n^3+119/6*n^2-65/6*n+1], [[1, 10] , 27/8*n^4-63/4*n^3+193/8*n^2-51/4*n+1], [[1, 11], 25/6*n^4-115/6*n^3+173/6*n^2 -89/6*n+1], [[2, 3], 1/6*n^4-n^3+7/3*n^2-5/2*n+1], [[2, 5], 3/2*n^4-5*n^3+13/2* n^2-4*n+1], [[2, 7], 25/6*n^4-35/3*n^3+37/3*n^2-35/6*n+1], [[2, 9], 49/6*n^4-21 *n^3+119/6*n^2-8*n+1], [[2, 11], 27/2*n^4-33*n^3+29*n^2-21/2*n+1], [[3, 4], 3/8 *n^4-7/4*n^3+25/8*n^2-11/4*n+1], [[3, 5], 3/2*n^4-9/2*n^3+6*n^2-4*n+1], [[3, 7] , 6*n^4-13*n^3+11*n^2-5*n+1], [[3, 8], 75/8*n^4-75/4*n^3+129/8*n^2-27/4*n+1], [ [3, 10], 147/8*n^4-133/4*n^3+181/8*n^2-31/4*n+1], [[3, 11], 24*n^4-42*n^3+30*n^ 2-10*n+1], [[4, 5], 2/3*n^4-8/3*n^3+23/6*n^2-17/6*n+1], [[4, 7], 6*n^4-12*n^3+ 12*n^2-6*n+1], [[4, 9], 50/3*n^4-80/3*n^3+95/6*n^2-35/6*n+1], [[4, 11], 98/3*n^ 4-140/3*n^3+94/3*n^2-31/3*n+1], [[5, 6], 25/24*n^4-15/4*n^3+107/24*n^2-11/4*n+1 ], [[5, 7], 25/6*n^4-55/6*n^3+53/6*n^2-29/6*n+1], [[5, 8], 75/8*n^4-65/4*n^3+ 101/8*n^2-23/4*n+1], [[5, 9], 50/3*n^4-25*n^3+125/6*n^2-17/2*n+1], [[5, 11], 75 /2*n^4-95/2*n^3+41/2*n^2-13/2*n+1], [[6, 7], 3/2*n^4-5*n^3+5*n^2-5/2*n+1], [[6, 11], 75/2*n^4-45*n^3+33*n^2-23/2*n+1], [[7, 8], 49/24*n^4-77/12*n^3+131/24*n^2-\ 25/12*n+1], [[7, 9], 49/6*n^4-91/6*n^3+34/3*n^2-16/3*n+1], [[7, 10], 147/8*n^4-\ 105/4*n^3+149/8*n^2-31/4*n+1], [[7, 11], 98/3*n^4-119/3*n^3+61/3*n^2-22/3*n+1], [[8, 9], 8/3*n^4-8*n^3+35/6*n^2-3/2*n+1], [[8, 11], 24*n^4-32*n^3+18*n^2-7*n+1] , [[9, 10], 27/8*n^4-39/4*n^3+49/8*n^2-3/4*n+1], [[9, 11], 27/2*n^4-45/2*n^3+27 /2*n^2-11/2*n+1], [[10, 11], 25/6*n^4-35/3*n^3+19/3*n^2+1/6*n+1]] ------------------------------------------------------------ This ends this article that took, 9551.036, seconds. to generate.