W Code - Pattern-Based DSA Learning Platform

Bhanu Bisht

Probability Theory

Reasoning under uncertainty — the language of ML predictions

Topics

🇮🇳 Indian Context

Spam detection for Indian emails, Zomato order prediction, and exam pass rates!

Probability Fundamentals

Foundation

Python Implementation

import numpy as np

print("=" * 55)
print("PROBABILITY FUNDAMENTALS")
print("=" * 55)

# Coin toss experiment
np.random.seed(42)
n_flips = 10000
flips = np.random.choice(['H', 'T'], size=n_flips)
p_heads = np.mean(flips == 'H')
print(f"\nCoin toss ({n_flips} flips):")
print(f"  P(Heads) ≈ {p_heads:.4f} (theory: 0.5)")

# Dice example
rolls = np.random.randint(1, 7, size=n_flips)
p_six = np.mean(rolls == 6)
p_even = np.mean(rolls % 2 == 0)
print(f"\nDice ({n_flips} rolls):")
print(f"  P(6) ≈ {p_six:.4f} (theory: {1/6:.4f})")
print(f"  P(even) ≈ {p_even:.4f} (theory: 0.5)")

# Conditional Probability
print("\n" + "=" * 55)
print("CONDITIONAL PROBABILITY")
print("=" * 55)

# P(pass | studied) vs P(pass | not studied)
# 100 students: 60 studied, 40 didn't
# Of those who studied: 50 passed
# Of those who didn't: 10 passed
total = 100
studied = 60; not_studied = 40
passed_studied = 50; passed_not_studied = 10

p_pass_given_study = passed_studied / studied
p_pass_given_no_study = passed_not_studied / not_studied
p_pass = (passed_studied + passed_not_studied) / total

print(f"\n📊 Indian Example: Exam Results")
print(f"  P(Pass | Studied)     = {passed_studied}/{studied} = {p_pass_given_study:.3f}")
print(f"  P(Pass | Not Studied) = {passed_not_studied}/{not_studied} = {p_pass_given_no_study:.3f}")
print(f"  P(Pass)               = {passed_studied + passed_not_studied}/{total} = {p_pass:.3f}")

# Independence Check
print(f"\n  Independent? P(Pass,Study) = P(Pass)×P(Study)?")
p_ps = passed_studied / total  # Joint
p_p = p_pass  # Marginal
p_s = studied / total
print(f"  P(Pass ∩ Study) = {p_ps:.3f}")
print(f"  P(Pass) × P(Study) = {p_p:.3f} × {p_s:.3f} = {p_p*p_s:.3f}")
print(f"  Independent? {abs(p_ps - p_p*p_s) < 0.01} → {'Yes' if abs(p_ps - p_p*p_s) < 0.01 else 'No, studying matters!'}")

Key Takeaways

P(A) is always between 0 and 1; P(Ω) = 1

Conditional probability: P(A|B) = P(A∩B)/P(B)

Independence: P(A∩B) = P(A)×P(B) means events don't affect each other

Every ML classifier outputs conditional probabilities: P(class|features)

P(Aᶜ) = 1 - P(A) is used everywhere in binary classification

import numpy as np print("=" * 55) print("PROBABILITY FUNDAMENTALS") print("=" * 55) # Coin toss experiment np.random.seed(42) n_flips = 10000 flips = np.random.choice(['H', 'T'], size=n_flips) p_heads = np.mean(flips == 'H') print(f"\nCoin toss ({n_flips} flips):") print(f" P(Heads) ≈ {p_heads:.4f} (theory: 0.5)") # Dice example rolls = np.random.randint(1, 7, size=n_flips) p_six = np.mean(rolls == 6) p_even = np.mean(rolls % 2 == 0) print(f"\nDice ({n_flips} rolls):") print(f" P(6) ≈ {p_six:.4f} (theory: {1/6:.4f})") print(f" P(even) ≈ {p_even:.4f} (theory: 0.5)") # Conditional Probability print("\n" + "=" * 55) print("CONDITIONAL PROBABILITY") print("=" * 55) # P(pass | studied) vs P(pass | not studied) # 100 students: 60 studied, 40 didn't # Of those who studied: 50 passed # Of those who didn't: 10 passed total = 100 studied = 60; not_studied = 40 passed_studied = 50; passed_not_studied = 10 p_pass_given_study = passed_studied / studied p_pass_given_no_study = passed_not_studied / not_studied p_pass = (passed_studied + passed_not_studied) / total print(f"\n📊 Indian Example: Exam Results") print(f" P(Pass | Studied) = {passed_studied}/{studied} = {p_pass_given_study:.3f}") print(f" P(Pass | Not Studied) = {passed_not_studied}/{not_studied} = {p_pass_given_no_study:.3f}") print(f" P(Pass) = {passed_studied + passed_not_studied}/{total} = {p_pass:.3f}") # Independence Check print(f"\n Independent? P(Pass,Study) = P(Pass)×P(Study)?") p_ps = passed_studied / total # Joint p_p = p_pass # Marginal p_s = studied / total print(f" P(Pass ∩ Study) = {p_ps:.3f}") print(f" P(Pass) × P(Study) = {p_p:.3f} × {p_s:.3f} = {p_p*p_s:.3f}") print(f" Independent? {abs(p_ps - p_p*p_s) < 0.01} → {'Yes' if abs(p_ps - p_p*p_s) < 0.01 else 'No, studying matters!'}")