(a) majority-vote accuracy (simulated) and the upper bound on its error; '-' where the bound does not apply
case                                       N=1           N=5          N=15          N=45         N=135         N=405
A: p0=0.30, 9 wrong at 0.078      0.300/ 1.000  0.458/ 1.000  0.738/ 1.000  0.970/ 0.308  1.000/ 0.000  1.000/ 0.000
B: (0.40, 0.50, 0.10)             0.396/   nan  0.387/   nan  0.332/   nan  0.238/   nan  0.110/   nan  0.015/   nan
C: (0.55, 0.45)                   0.550/ 0.995  0.592/ 0.975  0.659/ 0.927  0.748/ 0.798  0.877/ 0.507  0.977/ 0.131
D: (0.45, 0.55)                   0.446/   nan  0.409/   nan  0.348/   nan  0.253/   nan  0.126/   nan  0.023/   nan
   (each cell: accuracy / bound on error)

(b) population of 2000 prompts, K = 10 answers, answer distribution (1-b) Dirichlet(0.3,...,0.3) + b e_0, b ~ Uniform(0, 0.8)
single-sample accuracy 0.4638;  fraction of prompts where the correct answer is the unique mode 0.6965;  fraction with p0 > 0 1.0000
     N   pass@N  majority@N
     1   0.4750      0.4750
     5   0.8485      0.5735
    15   0.9610      0.6450
    45   0.9875      0.6645
   135   0.9975      0.6950
   405   0.9995      0.6885
