Example: White Token Out Of A Bag

Example: White Token Out Of A Bag#

(MacKay Exercise 3.12)

A bag contains one token, known to be either white or black. A white token is put in, the bag is shaken, and a token is drawn out, which proves to be white. What is now the chance of drawing another white token?

Analytical Solution#

Given that we have picked a white token, the probability that it had already contained another white, thus after adding the white it became WW:

\[P(WW|W) = \frac{P(W|WW).P(WW)}{P(W)} = \frac{1\times1/2}{P(W)}=\frac{1/2}{P(W)}\]

and the probability that it had a black token beforehand, so after adding the white it became WB is:

\[P(WB|W) = \frac{P(W|WB).P(WB)}{P(W)} = \frac{1/2\times1/2}{P(W)}=\frac{1/4}{P(W)}\]

We can evaluate the \(P(W)\) in the denominator via two ways:

  • By normalization:
    Since there is no other possibility, \(P(WW|W) + P(WS|W)\) should be equal to 1. Hence:

\[\frac{1/2}{P(W)} + \frac{1/4}{P(W)} = \frac{3/4}{P(W)} = 1\Rightarrow P(W)=\frac{3}{4}\]

  • By inference:
    As \(P(A) = \sum_i{P(A|B_i)P(B_i)}\)
    So, expanding and summing over all the possible cases (WW and WB) yield:

\[P(W) = P(W|WW).P(WW) + P(W|WB).P(WB) = (1\times1/2) + (1/2\times1/2)=\frac{3}{4}\]

Substituting \(P(W)\) in the above equations:

\[\boxed{P(WW|W)=\frac{1/2}{P(W)}=\frac{2}{3}}\]
\[P(WB|W)=\frac{1/4}{P(W)}=\frac{1}{3}\]

Monte Carlo Solution#

import numpy as np

N = 10000000

A = np.random.randint(0,2,(N,3))
# 0 : Black
# 1 : White

A[:,1] = 1
A[:,2] = 999

A[A[:,0] == A[:,1],2] = 1 # If both are white, outcome will be white
B = A.copy()

# If one is black, one is white, the outcome can be either
B[B[:,2]==999,2] = np.random.randint(0,2,np.sum(B[:,2]==999))

# Filter out the outcomes with white:
W = B[B[:,2] == 1]

num_TOT = W.shape[0]
num_W = np.sum(W[:,0] == 1)
num_B = num_TOT - num_W
print("White: {:d}/{:d} ({:5f})\t|\tBlack: {:d}/{:d} ({:5f})".format(num_W,num_TOT,
                                                                     num_W/num_TOT,
                                                                     num_B,num_TOT,
                                                                     num_B/num_TOT))

White: 5000604/7500747 (0.666681)	|	Black: 2500143/7500747 (0.333319)
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