Explicit Expressions for the Exact Number of Extended Simpson Paradox Scenarios By Shalosh B. Ekhad . Definition: For a and b posivie integers, let, A[a, b](n), be the number of Extended Simpson Paradox scenarios 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 2 A[1, 2](n) = 1/24 n - 1/12 n - 1/24 n + 1/12 n and in Maple format A[1,2](n) = 1/24*n^4-1/12*n^3-1/24*n^2+1/12*n 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 + 1/6 n - 1/6 n - 1/6 n and in Maple format A[1,3](n) = 1/6*n^4+1/6*n^3-1/6*n^2-1/6*n 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 + 3/4 n + 1/8 n - 1/4 n and in Maple format A[1,4](n) = 3/8*n^4+3/4*n^3+1/8*n^2-1/4*n 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 + 5/3 n + 5/6 n - 1/6 n and in Maple format A[1,5](n) = 2/3*n^4+5/3*n^3+5/6*n^2-1/6*n 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 35 3 47 2 A[1, 6](n) = -- n + -- n + -- n + 1/12 n 24 12 24 and in Maple format A[1,6](n) = 25/24*n^4+35/12*n^3+47/24*n^2+1/12*n 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 + 9/2 n + 7/2 n + 1/2 n and in Maple format A[1,7](n) = 3/2*n^4+9/2*n^3+7/2*n^2+1/2*n 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 4 3 2 A[2, 3](n) = 1/6 n - 1/3 n - 1/6 n + 1/3 n and in Maple format A[2,3](n) = 1/6*n^4-1/3*n^3-1/6*n^2+1/3*n 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, 8 4 3 2 A[2, 5](n) = 3/2 n + n - n - 1/2 n and in Maple format A[2,5](n) = 3/2*n^4+n^3-n^2-1/2*n 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, 9 4 3 2 A[2, 7](n) = 25/6 n + 5 n - 1/6 n - n and in Maple format A[2,7](n) = 25/6*n^4+5*n^3-1/6*n^2-n 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, 10 4 3 2 A[3, 4](n) = 3/8 n - 3/4 n - 3/8 n + 3/4 n and in Maple format A[3,4](n) = 3/8*n^4-3/4*n^3-3/8*n^2+3/4*n 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, 11 4 3 2 A[3, 5](n) = 3/2 n - 1/2 n - n and in Maple format A[3,5](n) = 3/2*n^4-1/2*n^3-n^2 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, 12 4 3 2 A[3, 7](n) = 6 n + 3 n - 3 n - n and in Maple format A[3,7](n) = 6*n^4+3*n^3-3*n^2-n 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, 13 4 3 2 A[4, 5](n) = 2/3 n - 4/3 n - 2/3 n + 4/3 n and in Maple format A[4,5](n) = 2/3*n^4-4/3*n^3-2/3*n^2+4/3*n 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, 14 4 2 A[4, 7](n) = 6 n - 3/2 n - 1/2 n and in Maple format A[4,7](n) = 6*n^4-3/2*n^2-1/2*n 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, 15 25 4 25 3 25 2 25 A[5, 6](n) = -- n - -- n - -- n + -- n 24 12 24 12 and in Maple format A[5,6](n) = 25/24*n^4-25/12*n^3-25/24*n^2+25/12*n 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, 16 4 3 2 A[5, 7](n) = 25/6 n - 5/2 n - 13/6 n + 1/2 n and in Maple format A[5,7](n) = 25/6*n^4-5/2*n^3-13/6*n^2+1/2*n 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, 17 4 3 2 A[6, 7](n) = 3/2 n - 3 n - 3/2 n + 3 n and in Maple format A[6,7](n) = 3/2*n^4-3*n^3-3/2*n^2+3*n 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 To sum up here is the data in Maple format [[[1, 2], 1/24*n^4-1/12*n^3-1/24*n^2+1/12*n], [[1, 3], 1/6*n^4+1/6*n^3-1/6*n^2-\ 1/6*n], [[1, 4], 3/8*n^4+3/4*n^3+1/8*n^2-1/4*n], [[1, 5], 2/3*n^4+5/3*n^3+5/6*n ^2-1/6*n], [[1, 6], 25/24*n^4+35/12*n^3+47/24*n^2+1/12*n], [[1, 7], 3/2*n^4+9/2 *n^3+7/2*n^2+1/2*n], [[2, 3], 1/6*n^4-1/3*n^3-1/6*n^2+1/3*n], [[2, 5], 3/2*n^4+ n^3-n^2-1/2*n], [[2, 7], 25/6*n^4+5*n^3-1/6*n^2-n], [[3, 4], 3/8*n^4-3/4*n^3-3/ 8*n^2+3/4*n], [[3, 5], 3/2*n^4-1/2*n^3-n^2], [[3, 7], 6*n^4+3*n^3-3*n^2-n], [[4 , 5], 2/3*n^4-4/3*n^3-2/3*n^2+4/3*n], [[4, 7], 6*n^4-3/2*n^2-1/2*n], [[5, 6], 25/24*n^4-25/12*n^3-25/24*n^2+25/12*n], [[5, 7], 25/6*n^4-5/2*n^3-13/6*n^2+1/2* n], [[6, 7], 3/2*n^4-3*n^3-3/2*n^2+3*n]] ------------------------------------------------------------ This ends this article that took, 158.577, seconds. to generate.