michal.piekarczyk.xyz

Here, looking at the recent recount in Georgia as an example around determining, what would a good number of votes to randomly sample to help validate an election result.

According to the latest results of this writing there is

B, T = 2472182, 2458010
p_observed = B*1./(B + T) # 0.501437

For now just starting off below visualizing the binomial distribution around the p_observed for different numbers of samples to look at what likelihoods are assigned to the different outcomes specified by that given binomial distribution.

from scipy.stats import binom
import numpy as np
import matplotlib.pyplot as plt
import pylab

B, T = 2472182, 2458010
p_observed = B*1./(B + T)


def do_plot(n, p):
    fig, ax = plt.subplots(1, 1)
    fig.patch.set_facecolor('xkcd:mint green')

    mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
#    print('mean, ', mean, 'var', var)
    print('mean, ', mean/n, 
          #'var', var/n
         )
    x = np.arange(binom.ppf(0.01, n, p),

                  binom.ppf(0.99, n, p))
    y = binom.pmf(x, n, p)
    print('sum y,', y.sum())
    # with plt.style.context('fivethirtyeight'):
    x2 = x/n
    ax.plot(x2, y, 'bo', ms=8, label='binom pmf')
    ax.vlines(x2, 0, y, colors='b', lw=5, alpha=0.5)
    plt.show()
    return list(zip(x2, y))

p=p_observed
print(f'n={n}, p={p}')
a = do_plot(n=10, p=p)
sum(x[1] for x in a if x[0] < 0.5)

n=10, p=0.5014372665405323
mean,  0.5014372665405323
sum y, 0.9880537609732434

png

0.37247237316599874

p=p_observed
print(f'n={n}, p={p}')
a = do_plot(n=100, p=p)
sum(x[1] for x in a if x[0] < 0.5)

n=10, p=0.5014372665405323
mean,  0.5014372665405323
sum y, 0.9789685520962167

png

0.4390661773494172

p=p_observed
print(f'n={n}, p={p}')
a = do_plot(n=1000, p=p)
sum(x[1] for x in a if x[0] < 0.5)

n=10, p=0.5014372665405323
mean,  0.5014372665405323
sum y, 0.9790245952379818

png

0.44151098858308324

p=p_observed
print(f'n={n}, p={p}')
do_plot(n=10000, p=p)

n=10, p=0.5014372665405323
mean,  0.5014372665405323
sum y, 0.9801955037573005

png

p=p_observed
print(f'n={n}, p={p}')
do_plot(n=100000, p=p)

n=10, p=0.5014372665405323
mean,  0.5014372665405323
sum y, 0.9800587882139997

png

Seeing above, that