Bertrand's Box Posterior

Problem

Each box holds items described by a [gold, silver] count pair, and a uniformly chosen box yields a gold item. Return the posterior probability that the next item drawn from the same box is also gold, weighting boxes by the probability g / (g + s) of producing the observed gold:

P = sum over boxes of w * (g - 1) / (g + s - 1) divided by sum of w, with w = g / (g + s).

Return 0.0 when no gold can be drawn.

Hint

Condition on which box was chosen; boxes with more gold items are more likely to have produced the draw.

Test cases

case 1

in: [[[2,0],[0,2],[1,1]]]

expected: 0.6666666666666666

case 2

in: [[[1,0],[0,1]]]

expected: 0

case 3

in: [[[1,1]]]

expected: 0

case 4

in: [[[3,1],[0,2]]]

expected: 0.6666666666666666

case 5

in: [[]]

expected: 0

Starter codePython

def bertrand_box_posterior(boxes):
    # Your code here
    pass

5 test cases · run your code against them in the editor