   n   k      p     true | E[unbiased]      sd | E[plug-in]      sd     bias
  16   4   0.01   0.0394 |      0.0394  0.0967 |     0.0359  0.0882  -0.0035
  16   4   0.05   0.1855 |      0.1855  0.1883 |     0.1700  0.1737  -0.0155
  16   4    0.2   0.5904 |      0.5904  0.2125 |     0.5529  0.2056  -0.0375
  16   4    0.5   0.9375 |      0.9375  0.0721 |     0.9134  0.0816  -0.0241
  16   8   0.01   0.0773 |      0.0773  0.1870 |     0.0628  0.1525  -0.0145
  16   8   0.05   0.3366 |      0.3366  0.3178 |     0.2810  0.2715  -0.0556
  16   8    0.2   0.8322 |      0.8322  0.2060 |     0.7578  0.2139  -0.0744
  16   8    0.5   0.9961 |      0.9961  0.0173 |     0.9858  0.0309  -0.0103
  64  16   0.01   0.1485 |      0.1485  0.1727 |     0.1334  0.1562  -0.0151
  64  16   0.05   0.5599 |      0.5599  0.2117 |     0.5178  0.2036  -0.0420
  64  16    0.2   0.9719 |      0.9719  0.0355 |     0.9571  0.0445  -0.0148
  64  16    0.5   1.0000 |      1.0000  0.0001 |     0.9999  0.0002  -0.0001
  64  32   0.01   0.2750 |      0.2750  0.3030 |     0.2247  0.2524  -0.0504
  64  32   0.05   0.8063 |      0.8063  0.2195 |     0.7261  0.2244  -0.0802
  64  32    0.2   0.9992 |      0.9992  0.0049 |     0.9962  0.0110  -0.0030
  64  32    0.5   1.0000 |      1.0000  0.0000 |     1.0000  0.0000  -0.0000

simulation (n=16, k=8, p=0.05): mean unbiased 0.3359, mean plug-in 0.2803, true 0.3366

population of prompts: p_x ~ Beta(0.3, 1.2) (many hard prompts), n = 16 samples per prompt, k = 8
  true mean pass@8 = 0.4962
   prompts M   sd of estimate (simulated)   between-prompt var/M  within-prompt var/M
          50                       0.0673                 0.0569               0.0264
         200                       0.0311                 0.0285               0.0132
        1000                       0.0148                 0.0127               0.0059
