Probability Basics

Implement fundamental probability calculations including conditional probability and Bayes' theorem.

Functions to implement

1. joint_probability(p_a, p_b)

Compute P(A and B) assuming A and B are independent.

• Input: P(A) and P(B)
• Output: P(A) * P(B)

2. conditional_probability(p_a_and_b, p_b)

Compute P(A|B) = P(A and B) / P(B).

• Input: P(A and B), P(B)
• Output: Conditional probability

3. bayes_theorem(p_b_given_a, p_a, p_b)

Compute P(A|B) using Bayes' theorem.

• Input: P(B|A), P(A), P(B)
• Output: P(A|B)

4. total_probability(p_b_given_a, p_a, p_b_given_not_a)

Compute P(B) using the law of total probability.

• Input: P(B|A), P(A), P(B|not A)
• Output: P(B)

5. bayesian_update(prior, likelihood, evidence)

Perform a Bayesian update to compute posterior.

• Input: Prior P(H), Likelihood P(E|H), Evidence P(E)
• Output: Posterior P(H|E)

Examples

# Independent events
joint_probability(0.5, 0.3)  # 0.15

# Conditional probability
conditional_probability(0.15, 0.3)  # 0.5

# Bayes theorem
bayes_theorem(0.8, 0.1, 0.14)  # ~0.571

# Classic disease example
# P(disease) = 0.01
# P(positive | disease) = 0.95
# P(positive | no disease) = 0.05
p_b = total_probability(0.95, 0.01, 0.05)  # 0.0594
p_disease_given_positive = bayes_theorem(0.95, 0.01, p_b)  # ~0.16

Notes

• All probabilities should be between 0 and 1
• Handle edge cases (e.g., division by zero)