Summary  
This chapter shows how to implement code to calculate conditional probability and Bayes’ theorem—computing joint, marginal, and posterior probabilities—and printing the results.

General domain of usage  
Medical testing

## Conditional Probability

Conditional probability measures the chance of an event happening given another event has already occurred.

**Formula:**

$$
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
$$

P_A_and_B = 0.1  # Probability late AND raining
P_B = 0.2        # Probability raining

P_A_given_B = P_A_and_B / P_B
print(f"P(A|B) = {P_A_given_B:.2f}")  # Output: 0.5

**Interpretation:**
if it is raining, there's a 50% chance you will be late to work.

## Bayes' Theorem

Bayes' Theorem helps us find $P(A|B)$ when it's hard to measure directly, by relating it to $P(B|A)$ which is often easier to estimate.

**Formula:**

$$
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
$$

Where:

* $$P(A \mid B)$$ - probability of A given B (our goal);
* $$P(B \mid A)$$ - probability of B given A;
* $$P(A)$$ - prior probability of A;
* $$P(B)$$ - total probability of B.

**Expanding $$P(B)$$**

$$
P(B) = P(B \mid A) P(A) + P(B \mid \neg A) P(\neg A)
$$

P_A = 0.01                # Disease prevalence
P_not_A = 1 - P_A
P_B_given_A = 0.99        # Sensitivity
P_B_given_not_A = 0.05    # False positive rate

# Total probability of testing positive
P_B = (P_B_given_A * P_A) + (P_B_given_not_A * P_not_A)
print(f"P(B) = {P_B:.4f}")  # Output: 0.0594

# Apply Bayes’ Theorem
P_A_given_B = (P_B_given_A * P_A) / P_B
print(f"P(A|B) = {P_A_given_B:.4f}")  # Output: 0.1672

**Interpretation:**
Even if you test positive, there is only about a 16.7% chance you actually have the disease.

## Key Takeaways

* Conditional probability finds the chance of A happening when we know B occurred;
* Bayes' Theorem flips conditional probabilities to update beliefs when direct measurement is difficult;
* Both are essential in data science, medical testing, and machine learning.

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Implementing Conditional Probability & Bayes' Theorem in Python

Conditional Probability

Bayes' Theorem

Key Takeaways