⚖️ Bayes' Theorem

Bayes' Theorem is the mathematical formula for updating our beliefs when we see new evidence.

🩺 The Doctor's Dilemma

Imagine a test for a rare disease (1 in 10,000 have it). The test is 99% accurate. You test positive. Are you doomed? No! Because the disease is so rare, a positive test is actually more likely to be a false positive than a true infection!

🐍 Python Implementation

Let's write a simple Bayesian updater function.

def bayes_theorem(p_disease, p_positive_given_disease, p_positive_given_no_disease):
    # P(Disease)
    prior = p_disease
    # P(Positive | Disease)
    true_positive = p_positive_given_disease
    # P(Positive | No Disease)
    false_positive = p_positive_given_no_disease
    
    # P(Positive overall) = (P(D)*P(+|D)) + (P(No D)*P(+|No D))
    p_positive = (prior * true_positive) + ((1 - prior) * false_positive)
    
    # P(Disease | Positive) = (P(+|D) * P(D)) / P(+)
    posterior = (true_positive * prior) / p_positive
    return posterior

chance = bayes_theorem(p_disease=0.0001, p_positive_given_disease=0.99, p_positive_given_no_disease=0.01)
print(f"Chance you actually have the disease: {chance * 100:.2f}%")