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.

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Implementing Conditional Probability & Bayes' Theorem in Python

Conditional Probability

Bayes' Theorem

Key Takeaways