bayes.py

gistfile1.txt

#simple implementation of https://en.wikipedia.org/wiki/Bayes%27_theorem

def bayes(p_a, p_b, p_b_a):
    """
        arguments:
        p_a: P(A), the probability of A
        p_b: P(B), the probability of B
        p_b_a: P(B|A), the probability of B, given A
        returns P(A|B), the probability of A, given B
    """
    return p_b_a * p_a / p_b

def coin_problem(x, y, z):
    """
    Given:
    1) all counterfeit coins come up heads with probability X.
    2) all genuine coins come up heads with probability Y.
    3) a coin you find on the ground has probability Z of being counterfeit. Equivalently, a coin is genuine with probability 1-Z.
    4) You find a coin on the ground and flip it once. The result is heads.

    Result:
    The probability that your coin is counterfeit.
    """

    #let A be "the coin is counterfeit"
    #let B be "you flip a coin once and it comes up heads"
    p_b_a = x
    p_a = z
    
    #formula derived by magic.
    p_b = x*z + y*(1-z)
    p_a_b = bayes(p_a, p_b, p_b_a)
    return p_a_b

test_cases = [
    (0.75, 0.5, 0.1,   "typical case"),
    (0.75, 0.5, 0.9,   "the counterfeiter made a whole lot of coins"),
    (1,    0.5, 0.1,   "the counterfeiter makes only double-headed coins"),
    (0,    0.5, 0.1,   "the counterfeiter makes only double-tailed coins"),
    (0.75, 0.75, 0.1,  "the mint isn't very good at balancing coins"),
    (0.5,  0.75, 0.1,  "counterfeiter wants to show them how it's done")
]

for x,y,z, commentary in test_cases:
    print(f"coin_problem({x:.2f}, {y:.2f}, {z:.2f}) == {coin_problem(x,y,z):.4f} \t {commentary}")

"""
output:
coin_problem(0.75, 0.50, 0.10) == 0.1429         typical case
coin_problem(0.75, 0.50, 0.90) == 0.9310         the counterfeiter made a whole lot of coins
coin_problem(1.00, 0.50, 0.10) == 0.1818         the counterfeiter makes only double-headed coins
coin_problem(0.00, 0.50, 0.10) == 0.0000         the counterfeiter makes only double-tailed coins
coin_problem(0.75, 0.75, 0.10) == 0.1000         the mint isn't very good at balancing coins
coin_problem(0.50, 0.75, 0.10) == 0.0690         counterfeiter wants to show them how it's done
"""